Yeah - you're not going to get anywhere you can guarantee politically to not get locked out of if something like the pandemic repeats. But you can do yourselves favours by being fully vaxxed etc. Maybe somewhere like Bosnia- Herzogovina or Bulgaria might be more relaxed.
Value for money wise you can probably get most in Italy followed in order by France, Austria and Switzerland of the main alpine nations. Places like Slovenia and Southern Poland you could probably get a lot.
Pyrenees likely to be better value I suspect and quite an interesting place to be where you wouldn't be just another loudmouthed yank
Seriously though France isn't specifically anti American. They are suspicious of all outsiders but worst of all Parisians.
Seasonpass wise everywhere has a seasonpass for between Eur600- 1000ish. Some are better than others with reciprocal resorts. Austria has a Tirolcars a Salzburg Superski and a really good value seasonpass for Innsbruck residents which includes Wacker FC, ice hockey and Schwimbads. Magic pass in CH is really cheap.
Well...we are the worst of two worlds in that case, as American expats who live in Paris. We bought an old farmhouse in a hamlet in the Haute-Savoie. Our 4 neighbors, all French, could not be nicer. Unless they spend their time talking about us behind our backs...having lived in France for 10 years and with dual citizenship, I think it's a little more complicated. French people are fundamentally conservative and quiet in public. They make their friends early; it's a small country where visiting with family and friends is a major activity. So they don't aggressively seek new friends in the way Americans do.
That said, there's also a city vs country divide. Parisians are busy people; work days are long (9-7 is normal); city life can be exhausting (metro-boulot-dodo). Many Parisians don't have time for niceties with tourists. But in the country the pace of life is a little slower and those who can communicate in English or Franglish are happy to banter and help you out. So, bottom line: If you learn some French and have a friendly and open attitude, you will be welcomed in the country/mountains. Same as anywhere.
As to the OP's question: Airport access is key. Geneva is a good hub for northern and central French Alps and some Swiss areas. Our place is within an hour of GVA, and even though we are not in a ski station, there is a small hill in our commune, but we are within an hour of 6 major stations: Portes des Soleil, Grand Massif, the Chamonix Valley, St.Gervais/Megeve/Contamines, La Clusaz/Grand Bornand and Espace Diamant. Aosta Valley is through the MB tunnel. So we'll never lack for variety.
Terrain/crowds: Just about any French station isn't crowded when you're not in the school holiday weeks. As for terrain: You are no doubt aware of the "off piste" situation, but won't go into that. Know that higher-altitude resorts are above the treeline so the tradeoff is better snow vs the chance of flat light/whiteouts. There are lots of mega-linked resorts in France so you'll never run out of new runs to try.
Costs: Window prices are way cheaper here; that said, I think the future is going to be multi-resort passes like Icon. Cost of living: France is cheaper than the US overall, with the exception of energy prices (we would be over the moon to pay $6 gallon for gas!). That said, I just paid a tax bill as a small business (i.e., myself) owner and it was like a dp on a house...and for expats much depends on forex rates.
I think it depends on what you want out of it. For us, our place is an escape from Paris in the short term and in the long-term, hopefully, we'll move there full time once our kids are out of the house in 4 years or so, and orient ourselves toward Geneva. It's big enough that we can host family and friends. It's in a really quiet, beautiful, peaceful area that is attractive in both summer and winter. in fact I think I like summer in the Alps even more than winter. Sounds like you're like us: we hike, cycle, climb so having those activities literally outside our front door (we live atop a col used often in the Tour de France) is an amazing privilige. But our commune has only one restaurant...
Much depends on your budget but you can get showplace properties in the French Alps for under $1 million if you don't need to be right on a ski lift.