Hyper Beam not recharging after KOing a Pokemon in RBY was a sick mechanic and should come back in some form, maybe nerfed slightly to curb the power increase through items and such. It encouraged knowing KO ranges in a way that's completely unlike any other move archetype in the series. The dynamics it introduced were super unique and gave a reward to skill-based gameplay. Switching in Pokemon to absorb the moves and force recharges was also a cool way to claw momentum back. I wouldn't call it centralising, but the way game states were manipulated was seriously an awesome thing. Imagine Rock Wrecker, Roar of Time and even the starter versions finally seeing use after so long. Hell, Tauros could replicate its old days with Giga Impact in lower tiers.
Maybe bring back that thing where recoil moves didn't damage you when KOing something too, that was only in Stadium but also really cool.
Definitely think all of this would become crazy with the general powercreep, depleted bulk and addition of items though. But if toned down, all of this stuff could be super interesting.
I think just letting the user completely skip the recharge next turn is a bit OP, but I do think they could maybe allow for a lesser version of the effect: If user knocks out the target, next turn user can use a Status Move that either targets themselves, an ally, or user's side of the field. If any other kind of Move is selected the player will be told this Pokemon needs to recharge this turn.
I think this combines the best of both ideas. When you select a Hyper Beam or one of its variants, you accept that for all this raw power you won't be able to do anything next turn. However, this is very turn-based logic and technically we're not supposed to be thinking that's how battles would be done IRL, so to make battles feel more fluid (and reward the player for a "precise" use of Hyper Beam) they could say that while the the opponent was selecting their next Pokemon the Hyper Beam user caught its breath enough for it to use a healing move, stat buffering move, doing something to help its ally, etc.. It's not ready to attack just yet but with the opposing trainer distracted for a moment it's not going to be counterattacked immediately.
However I don't think we need to really change recoil moves since knocking out the opponent with lower HP means the user receives less damage thus they're already awarded.