When Cersei asks Kevin to serve as Hand, he says he’s willing to do so only if she also gives him the regency. Beyond the desire to completely displace Cersei, does it make any sense for the regent and the Hand to be the same person?

I think the intent is to ensure clear lines of authority and to avoid political conflict (i.e, what happens when the Regent and the Hand disagree?). And we’ve seen Cersei conflict with Hands before, as she did with Tyrion. 

But we don’t have to stop there…look at the running conflict between  Aegon III’s Council of Regents and the Hand of the King Unwin Peake. There were lots of reasons for political instability during the False Dawn, but the conflict between Council and Hand was a major one.