It’s not entirely democratic at the convention, even in a typical election cycle. Recall that 15% of the delegates are “unpledged party leaders and elected officials” (superdelegates).
That aside, I don’t think there’s a constitutional issue here with respect to replacing Biden. If Dems decided to select the candidate via musical chairs or Parcheesi, to my knowledge that’s a party matter (though voters would undoubtedly take a dim view of such antics).