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Depends where you go in Europe, but my strong impression is that the relative cost of living is very significantly lower than FAANG-like equivalent locations in the US (SF, Seattle, NY, etc).

Remote for the US is definitely still the best way to get the benefit of the difference though, and even with a lower paying US role in return that could be worthwhile.

Remote work visas for Spain and I think Portugal are designed for exactly that case and should fit the bill nicely if you have remote work. Alternatively if you can't get any, it gets significantly more complicated - finding a well-off-relative-to-local-cost-of-living local software job is quite possible in many many major EU cities, but will take time and require some adjustment en route (to local norms, paperwork, and languages) so the visa situation can be challenging. Doable but requires research and work.