Is there a proof that shows why DFAs can't be used to show the concatenation of two regular languages?
Using NFAs does prove the result for regular languages, since NFAs accept exactly the regular languages. In principle, one could give a proof using DFAs but it's likely to be very fiddly and probably not enlightening. It's much easier to give the proof using NFAs and separately show that anything accepted by an NFA is also accepted by a DFA.

Indeed, it's even easier to give the proof using regular expressions! This is part of the point behind giving the three equivalent characterizations of the regular languages: it means that, when you want to prove something about regular languages, you can use whatever system makes the proof easiest.