"He and his brothers (or at least the next brother up from him, anyway) arranged that between themselves, though, I'd thought, rather than anyone else getting to "choose" exactly?"
A council of nobles got to choose the King, Alfred and his brothers probably did set the ball rolling and express a desire that certain things happened but that didn't make it so, and nor did just being the Monarch's eldest son.
I don't like the idea of a popularly elected monarch either, as you say we'd have some TV show come along and we'd end up with King Stavros Flattley or whoever else is the short lived flavour of the month.
But the idea of perhaps getting the House of Lords to select a successor from the extended Royal Family could work. We might then be able to avoid Charles but get someone else not quite so divisive.
So let's roll back how it worked in the years before William the Conqueror and let's see Charles justify why on earth he should get the job :)