I can't think of any disadvantage to me gaining an additional European Union citizenship, with some exceptions (e.g. Estonia has military service).
Many millions of Europeans live in a different country to their citizenship. It's probably a lot more common than for US citizens, for all kinds of people (from the unemployed to millionaires), so the issues were solved long ago.
(I think there can be cases where if you don't live in a country "full time" you can lose the right to things like free healthcare, but that's separate.)