You don't need citizenship at all. Not even permanent or temporary residency. You will become a tax resident of a state even if you have just a tourist visa. You don't need to own the property either, a rental apartment is just fine. Even living in a hotel would work if you can prove it was your "base of operations" during that time.
Not sure what's the distinction between rich and wealthy, sounds the same to me.
Well I think I meant something in between. You don't need to be from old money to get a property in Monaco and much less in other places in EU - I got several, and I started literally from 0. And you definitely don't need to be from old money to rent an apartment in Monaco for half a year. A senior software engineer (contractor) could afford it and would profit from the operation if they live in Scandinavia/Germany/Austria/France.
I have a good enough tax regime where I live in Central Europe - for 2022 I paid 8% including income tax, health and social insurance. But I'd definitely consider doing this if it was over 20%.
Not sure what's the distinction between rich and wealthy, sounds the same to me.