From PHB 191:
"If a flying creature is knocked prone, has its speed reduced to 0, or is otherwise deprived of the ability to move, the creature falls, unless it has the ability to hover or it is being held aloft by magic."
From MM page 8:
"Some monsters have the ability to hover, which makes them hard to knock out of the air (as explained in the rules on flying in the PHB). Such a monster stops hovering when it dies."
That's the extent of the 5e rules on hovering as far as I can find. I never paid attention to flying rules in AD&D, so those hover rules you describe are news to me, but the 5e ones seem very reasonable. (Flying requires movement to stay aloft. Hovering is freaky anti-gravity.) The suggestion that flying creatures have to use their movement to stay aloft is not explicit in the rules, but is a logical and reasonable extrapolation. In my opinion. If players feel the need for more clearly codified rules and what they can and can't do while flapping their wings, the AD&D maneuverability classes seem reasonable to port over.
Personally, as a DM, I probably wouldn't bother with memorizing/keeping those rules handy for my flying NPCs, but the general spirit of them is still helpful.