On the issue of the yo-yo healing, some have suggested exhaustion levels when it happens, or long-term lingering injuries, or similar mechanics.
The main issue with those mechanics are that they're long term, and generally outside the scope of the immediate fight. They're also things that you can address "later". Most of the time they have limited, if any, effect on the ongoing fight.
I think a more immediate penalty would be more useful, and it has to be easily inserted into the existing 5E mechanics. Combat is the thing that is most easily slowed down and made worse by extra mechanics.
So the simple problem is: If a character drops to 0 HP, and is healed before his next turn, within the current combat system there is almost no downside. The only loss was the action of the player who healed the downed fighter, and that's something the healer was probably going to do anyway. Some classes can even do that as a bonus action, which means it doesn't really interrupt them at all.
The main issue on the fighter side is that when the fighter drops to 0, it doesn't happen on his turn; it's some sort of enemy attack. Assuming the healer heals him at the next available opportunity, about half the time it will get him back on his feet before his next turn, and half the time he'll lose one turn waiting for the heal.
So the yo-yo effect is most prominent when the fighter is downed, and then healed, between two adjacent turns, and when the healer doesn't have to spend any effort (action time) restoring the fighter to active ability. The fighter loses no combat time, and the healer loses no significant action time.
From that perspective, you want to penalize either the fighter's combat time, or the healer's action time. Changing the healer's action time is undesirable, as there are so many different ways to heal. Changing the fighter's combat time, however, is much easier. It's also more in line with players' expectations, as seen in the suggestions for exhaustion and the like.
So, from that perspective, I'd propose an extremely simple change that doesn't add new mechanics, and doesn't slow down combat: [[ If you are cured back up from 0 HP into the positives, you are stunned for 1 round. ]] The shock of being knocked out and being brought back from the brink of death is not so easily shaken off.
That means you're guaranteed the loss of one action round (aside from any rounds you were sitting at 0 HP), and can't just continue to attack uninterrupted. It also puts you in a vulnerable state during that time (attackers have advantage, you auto-fail Str and Dex saves, you can't move or act), which should certainly be the case after being taken down. It makes a party's position shakier when trying to recover from a dangerous situation.
This would not be a change to make things deadlier directly, but it's designed to address the issue that there is no consequence for dropping to 0 HP. So it prevents the most extreme form of yo-yo'ing, hopefully reducing the amount of aggravation it causes, and dialing back the need to kill characters just to prevent it.