Not all European GSM phones work in the US (regardless of the SIM card). In fact I tried to use one of my GSM flip phones (bought new in 2015) in New York and it just said "no network", it didn't even have the possibility for emergency calls.
Part of the relevant information is here on wikipedia: https://en.wikipedia.org/wiki/GSM_frequency_bands
Europe uses GSM 900 and 1800 ("dual band").
America (not only the US. Also Canada, South America, Central America) mostly uses GSM 850 and GSM 1900.
Smart phones are typically quad band phones. I assume the manufacturers don't want to bother with different models for different parts of the world, or nowadays, the possibility to use your smart phone everywhere is far more important than with "cheap" phones that could only text and phone - after all for these services you also need a decently priced plan, or might just be unavailable (time zone difference, vacation, …) or get a throw-away phone in the country you're visiting.
You already collected the relevant information about what GSM standards your phone can do, and in fact all relevant ones are covered.