One thing I did, and it didn't have a massive impact in the current game due to me not introducing it until very late, was to change long rests so that instead of automatically healing they had to spend hit dice.
It doesn't do much for the casters with their long rest spell regeneration, but I feel like it does make a short rest more appealing if the healing element is the exact same, because while I occasionally heard "I'm low on spells" as a reasoning, I much more often heard "We are injured, let's take a long rest" to the point where I had to fight one time to get the cleric to use Preserve Life and us simply short rest to spend our hit dice (we were full on HD) instead of long resting to heal our wounds.
We went into the next fight with nearly full hp and plenty of spells, and no long rest.
Also, thinking on my gaming experience, I rarely have problems with the "main questline" battles. It is the encounters I feel like should happen while traveling that give me headaches. The fights are meaningless, usually unconnected from the plot, and the players have potentially days or weeks to heal up. So, I either run multiple combats for every day they are traveling (my gods no, it would make it unbearable to try and get anywhere), accept a single encounter to waste everyone's time, or generally just skip them because there is no point to them anyways.
Still, I just think this is going to be something DMs are going to have to come to grips with and find solutions for. I liked some of the ideas Colville was proposing in the video. I just felt like A) they were too powerful [vulnerability to smite damage, holy crap on a stick] and B) They felt too arbitrary. I feel like I'd want a reason why they have these abilities that unlock in this manner. So, maybe I would use these rules in a special dungeon, where it is part of the dungeons nature to empower people who fight through it longer, but I don't think I would use the rules constantly.