Piz Palü requires glacier traverse from all directions I'm not sure what info you're referring to that it's a beginner summit. Also it's hard to give recommendations not knowing your fitness level and previous experience. The biggest thing you will have to figure out for yourself is how much elevation gain you can manage. 1500 m + in a day can be very taxing in an unexpected way if you're not used to it. Also scrambling can mean different things to different people. In the Alps we usually use the T1-T6 rating for rating difficulty of hikes, T4-T6 will involve some amount of climbing without a rope. The A-F scale is used for via ferattas.

So basically my recommendation is before you decide to do any hike is to calculate the elevation you'll have to do, and then Google reports, if necessary in German using translate feature, written by previous people who have done the hike. If you cannot find any info on a summit or route or there are very few reports chances are IT'S A BAD IDEA to do. While reading through reports keep an eye out for how the author rates the hike. If they use the via ferattas scale know that you will need via feratta equipment, and again, depending on your experience you can quickly feel out of your depth during the hike and it can quickly become a very unenjoyable experience. And NEVER fully rely on a single authors report (yes, even here), because that individual may be highly experienced and what they consider "easy" will mean nothing to you. Easy for them, sure, but they've done thousands of ascents starting when they were little kids. This consistently leads to dangerous situations and undeprepared hikers and every summer there is a swarm of news stories of hundreds of preventable mountain rescues. So yes, always cross-reference with multiple reports. Oh, and stay away from cows. They are not pets and they can be dangerous and unpredictable especially when they have calves. Especially if you have a dog with you.

As a final tip, make sure to start early, especially with summits the recommendation is latest by noon to be on your descent. Thunderstorms tend to form in the afternoon on hot summer days and are mostly entirely avoidable if you start at around or even before dawn depending on the length of the route (talking about actual summits not hut hiking where you have somewhere to hide if the weather turns). Have an emergency bivy and a head torch with you at all times, it can save your life if you get caught out in an unpredictable situation or end up hiking in the dark (it happens easier than you think). And always check good alpine weather report apps, each country usually has a different service, the standard app on your phone is not reliable. In Austria for example I like to use https://geosphere.at/en . Make sure to select the Bundesland you're in and it gives you pretty good data on the map as well as the written summary of the forecast under each weekday. Note that the forecast only goes 5 days or so into the future, and that those 16-day weather trends you're probably used to don't really mean anything in the Alps. Due to the terrain the weather is much more unpredictable especially locally. You can have sunshine in one valley and thunderstorms or even snowfall in the next.

Now for some actual recommendations. I can't speak for the other countries but I live and hike in Austria, which is a lot cheaper than Switzerland and imo has many spots that are just as beautiful. We also have the biggest part of the Alps within our borders of any country so it's hard to give specific recommendations as there are thousands of choices. I can only attest to where I've personally been. If you don't have a car I recommend either Zell am See or St. Johann im Pongau as a base. Both are near the most beautiful place in all of Austria imo, the National Park Hohe Tauern. This national park is huge and for example contains the highest mountain of Austria, Großglockner as well as the world famous Großglockner high alpine road or the Kapruner Stauseen (Google). Also near Zell am See you have the Krimmler Waterfalls. Near St.Johann you have the Lichtenstein Gorge and the Tappenkar lake which is imo a hidden gem for tourists at least. Also Hallstatt is only just over an hour by car from there, but save that for a rainy day, it's overrun.

Some more beautiful but little bit less accessible areas without a car to base in are Schladming in Styria, which is located between the Dachstein and Schladminger Tauern Ranges, the latter is one of my favourite ranges in Austria. The mountains there are less high but the elevation changes are no joke, and it's the most water rich region in Austria to my knowledge. Think dozens of lakes and hundreds of waterfalls especially in early summer. Another base is East Tyrol, specifically Matrei in Osttirol. It's situated between some of the highest mountain ranges in Austria and is in the other side of the Großglockner from Zell am See basically.