Unless you want to stay longer than allowed by a Schengen visa,
or if you want to open a local bank account,
or access local healthcare,
or
or if you want to open a local bank account,
or access local healthcare,
or
And first part of the OP's statement is true - its easy to retire into southern Europe from wealthier parts of Europe. Quality of life, safety, healthcare etc is usually more than fine.
So, practical question for Americans living in Europe: Are you able to just forgo local banks and use an internationally-minded US bank for everything? One without international fees? Can you just get your paychecks deposited there, say? Does this make anything impossible? Say, getting a mortgage?