The question in the title can be answered in many ways and all the answers might be correct. This all depends on what activity you are interested in when you go there.
The Alps are an area where I go every summer, and this implies mountaineering. So I feel I am qualified to say a few words about the Alps from this perspective.
But you can go there for skiing, for cycling, for hiking, for rock climbing, and for many other activities. Every part of this large mountain range has something to offer.
On the other hand, the terms like the best and the most beautiful are the matter of personal feeling and preference. Therefore there is no the best and the most beautiful area of the Alps that would be accepted as such for everybody.
However, here are some aspects to consider if you need some sort of answer to thee question in title:
- The highest concentration of 4000ers in the Alps is around Zermatt in Swiss Alps.
- The highest elevation in the Alps is Mont Blanc, so this is the border between Italy and France, and you cannot put it in one country only.
- The greatest mountain passes are in the Italian Alps.
For all other aspects, please read more in the text below.

The Alps for skiing
No matter where you go to the Alps, you will have plenty of options for skiing. Many of lifts work also in summer time for tourists and for mountaineers.
But I would point out at least two areas in the Alps where you can enjoy skiing throughout the year.
One is at Stelvio pass where there are plenty of ski runs to ski in summer. This area is particularly interesting also because of the Stelvio road (there is a picture farther in the text) which is one of the greatest passes in the world. But this is also a paradise for hiking, mountaineering, and cycling.

The Pass itself is at a very high elevation of 2757 m, but it is surrounded with many peaks that are well over 3000 meters high. The picture above shows how it was on my tour to Punta degli Spiriti (3467 m) which is the peak in the middle. There are many options on the Pass for accommodation.
Yet another similar place where you can ski in summer is Breithorn Plateau with an elevation of around 3800 meters, close to Breithorn, in the Zertmatt area in Swiss Alps. The picture below is from my tour to Breithorn and it is taken in summer time. There were many skiers around.

This environment is very different from the mentioned Stelvio Pass area because it is higher and there is no road. So this is a high plateau where you get with gondola from Zermatt and also in a similar way from the Italian side.
The Alps for cycling
I would say that the most attractive areas for cycling are where you have high mountain passes. By this I mean road cycling of course.
So if this is your activity, I would warmly suggest the Stelvio mentioned above, but also Gavia Pass which is not far in the same part of the Italian Alps. The picture below shows how it looks at Stelvio.

Yet another area which I would put forward for cycling is Dolomites. There are several reasons for this. First, this is again an area with many beautiful passes. They are lower as compared with those mentioned above, but there are many of them, and in an incredibly beautiful surrounding.
Yet another reason for suggesting Dolomites is that you can make round tours. I would point out the Sella Group with several passes around, and also with great roads that allow you to make round tours. For such a tour around the Sella mountains, you will go over four passes, Pordoi, Campolongo, Gardena, and Sella pass.
Off-road cycling in the Alps
For this activity you also have plenty of options. I would mention at least the following two.
Livigno area in the Italian Alps: Mountains around Livigno are high, the peaks are over 3000 meters, and Livigno itself is one of the highest cities in the Alps. It is at an elevation above 1800 meters, higher than Zermatt in Swiss Alps. But the mountains around are mild and green so you have endless options for using mountain bikes.
St. Moritz area in Swiss Alps: This is an area adjacent to Livigno, so the mountains are similar and you can pass from one to another country at many places. There are no border guards of course.
The Alps for hiking
Actually, it is very difficult to chose just a few areas out of so many. The term hiking may imply lots of activities as I discussed in my separate text about differences between hiking and mountaineering.
But if you are interested in endless walks with great and unobstructed views, I would again point out Livigno area. You can walk in any direction you want, you will be at high elevations, far above tree line so views are always guaranteed.
You can also get to the summits, see how Piz Casana (Piz Chaschauna, 3071 m) looks inviting, and this is Swiss-Italian border. You can get to the place where this picture is taken by a bicycle.

But there are no glaciers around Livigno, so if this is what you want to have in sight, it is best to go to Zermatt.
if you would rather stay in the shadow of trees, with occasional peaks and views, think about German Alps and the area around Alpspitze (see the top picture above) and Garmisch-Partenkirchen.
But if you want to go the highest elevations by your car, and then to make hiking tours around, this is then Stelvio Pass and the nearby Umbrail Pass before all.

The Alps for mountaineering
There are many possible aspects related to this activity. They include at least the following:
- Glacier tours.
- Demanding but snow-free summer-time tours.
- True alpinist tours.
Glacier tours in the Alps
There is no doubt Zermatt area is the first that comes to mind together with Mont Blanc of course. The highest and glaciated mountains are around Zermatt, so there is no other place in the Alps that could offer more.
But there are more of course. Much easier glacier tours you will find in the Monte Cevedale area in the Italian Alps, see the picture below. The picture is taken from my climb of Tresero (3594 m). There are several mountain chains that you can climb going from one peak to another. They are all below 4000 meters but that does not mean they are easy.

Note also that Austrian-Italian border goes over some of the highest mountains in that area. While the Italian side approach may be without glaciers, from the Austrian side you always have long glacier tours, see the picture below which shows the Austrian side of the chain.

Demanding but snow-free summer-time tours in the Alps
Many options are at your disposal. But here are a few things I want to point out. The highest snow-free mountain in the Alps in summer time is Barrhorn (3610 m) in Swiss Alps. This is just a walk up at high elevation.
Slovenian Alps are lower than those in the nearby countries, and they are snow free in summer. But they are hard because valleys are deep. Most of them can be climbed in one day, but there are many huts, their concentration is higher than in any other parts of the Alps. The picture below shows Triglav (2864 m) from my climb of Razor.

True alpinist tours in the Alps
You have great alpinist routes throughout the Alps, from the lowest to the highest mountains. Some of the mountains have extremely hard routes on one side and mild or even easy access routes from some other sides.
Most famous example of this type is Marmolada in the Italian Alps with its hard south face that has more than 200 alpinist routes. The access from the north is over a glacier but it is very mild. Something similar you have on Triglav with its steep north side and many easy access routes from the south side.
But there are mountains without a weak side, take Matterhorn as one obvious example. There are many books about Alps where you can read about its hardest routes and the highest mountains, some of the you can find on Amazon.
The Alps history tourism
Many areas in the Alps are with plenty of stuff from the WWI. This includes trenches, barb wire, man made tunnels and caves, shelters, and many other things. This holds in particular for Alto Adige (South Tirol) in the Italian Alps. If you go to Dolomites you will see plenty of it.
There is a very special place close to the Gavia Pass, where they collected the materials from military trenches and built a refuge. This is Bivacco Battaglione Ortles (3122 m). The picture below shows how it looked when I was there several years ago. It is always open, no guardian, there is a stove and beds with blankets.
This is high above tree lines, so there is no wood around, but there are plenty of planks from the war, after 120 years they are still good enough for fire. I made a fire by collecting them, there is an axe available for this purpose.

Did you hear about Hannibal crossing of the Alps with elephants in 218 BC? There is no consensus about the pass which he used. So go there and try to figure it out. I have my own theory about it.
Many books about history of Alps can be found on Amazon. If you rely on maps like me, then here you can find many of them for all areas of the Alps.
Huts, shelters & accommodation in the Alps
Regarding huts with guardians, I do not think that any area in the Alps is better than other. There are good and less good huts everywhere.
However, regarding shelters, I think you have them far more in the Italian Alps than anywhere else. They are usually simple metal constructions painted in red, always open, and usually with beds and blankets available. This shelter below Monte Confinale summit is a typical example, this is Bivacco del Piero at 3166 m of elevation.

Such shelters are rare in Swiss, Austrian and German Alps, but you have them in the Slovenian Alps though never in this color.
As for accommodation, I myself always book something through Booking.com, and they cover all the area, no difference in principle, so you can always find something suitable. Note that in summer time you can find very affordable studios in various ski centers in Swiss Alps and also in French Alps.
Conclusion
In summary, I do not think that there can be consensus about the best and the most beautiful area in the Alps by country or in any other sense.
But I do have my own preferences for some countries. So for example, in the Slovenian Alps, Kranjska Gora is the number one for me. I can only regret the fact that I have no permanent residence there.
As for Italy, well, I am in love with this country and I love Italians, they have a permanent and life-long credit from me. I feel great everywhere in Italy. If I would have to choose, perhaps this would be Livigno or Tre Cime di Lavaredo area. But have you ever visited Dolomites, in particular Corvara below Gardena pass? I could stay there forever.
Note that all the photos here in the text are my own from various tours, so let me know if you need details. Check also about drinking glacier meltwater in my separate text.
But how about some other areas? Have you ever visited Tenerife? If you plan to go there, you might want to climb El Teide, so read in my separate text about a climbing permit for this mountain.
This site is all about outdoors questions and answers and I add texts here on a regular basis, so bookmark it and keep as a reference. Please use the comment box below in the case of questions or comments. Thank you for reading and have a nice day.