I lived and worked abroad off and on (mostly on) from 2003 to 2018. I maintained my US-based bank and investment accounts and freely moved funds between them. Also moved funds between US bank and foreign banks, although today that may be easier to do with crypto currency. I used US addresses for my bank and investment activities. The only issues I had were sometimes dealing with dreadful foreign banks and their kooky rules and the occasional wire transfer fee.
Exactly. The key aspects are to 1) be a US citizen or permanent resident and 2) maintain a permanent US address. For those who cannot meet these prerequisites, their investment options into index funds and ETFs may be constrained.
The OP is an American already, so no issue there. Setting up a mailing address is trivial.
Yes, but... (#3)
I'm not sure that it is criminal to lie to your bank about where you live if you aren't committing fraud and you make sure to pay all of your taxes.
But yes, AFAIK US citizens in the EU are now in a world of hurt between UCITS, PRIIPS, and KID on the EU side and FATCA and PFIC taxation on the US side. Also, FATCA reporting is so onerous that plenty of brokerages just won't touch you if they find out your are a US citizen, even in non-EU countries.
I don't even think that a Schwab International or Interactive Brokers account can get you around this at this point, unless you lie and tell them that you live in some non-EU country. Unless perhaps you can get certified as a "professional" which would usually require half a million euros AFAIK.
EDITed to add: and I'm not sure that there is anything Vanguard could even do to make a US domiciled PRIIPS+KID compliant ETF as I'm pretty sure that UCITS requires that the fund be domiciled in the EU which makes it a PFIC.