in case you go for a cheap new phone, around here the best option are some Redmi ones. recently got a Redmi Note 9 and slapped LOS on it for a friend, for about 170€. since none are very recent models, sometimes I had to spend a good time searching for them in shop stocks
yes, they do (redmi the same as other xiaomi phones) and that's an important thing I left out.
I did create an account long ago with an alias email that i've used for the unlocking of several phones' bootloader. it also requires a phone number for "verification" when you start the unlocking process, so for anonimity you'd need an unregistered burner phone number