I agree with everyone else! I did GAS last fall and just wore my regular (Altra brand zero drop) athletic shoes for hiking in Switzerland. I did wind up leading my little band down a fairly narrow dirt trail into Muerren but it was fine in the footwear I had on. Hiking boots would have been overkill. I also walked Mannlichen to Kleine Scheidegg on Day 2 and it is level and wide. There are a few small rocks but not enough to need any kind of ankle support.
You'll have hiking opportunities on both days in Switzerland and the views are fabulous and jaw-dropping. You may have some short hiking options in Hallstatt (it was raining and I passed) and as mentioned, in Baden Baden. I had been to Neuschwanstein before so I walked part way around the lake which was lovely but if you've not visited, you'll probably want to go see the castle.
If you think you might want to do that Mozart concert in Salzburg (guide will probably get a head count and make reservations for you) there is a dress code. They don't allow athletic shoes. I skipped because that's all I had but if you think this will be of interest, take a pr of shies to wear there.
edited to take out misspellings due to fat fingers on a smartphone, lol!!
This is a really fun tour!