I guess what I am trying to fix is being 100% tip top shape overnight.
There's two different things here that you would need to figure out for yourself with regards to the success or failure of instituting your house rule-- the "in-world reality" you are trying to mimic, and the "characters having less mechanics strength" at the start of each "day".
If you are trying to mimic some in-world reality where you feel like a person should really take more than single day to naturally recover from their "injuries"... I can understand why you would want to try and put in a "slow healing" variant into your game. Players have been doing that since the beginning of 5E. Their belief that there should be some semblance (or at least a little nod) to verisimilitude when it comes to natural healing in D&D, so they find various rules and house rules to not allow for "overnight" or Long Rest full recovery. So in that regard, your spitballing of this rule is just as valid as any other house rule that has been invented. Now
personally... I don't think any of that actually accomplishes anything because all of the various "slow recovery" rule options that have been invented-- NONE of them even come close to mimicking any sort of reality in how long it actually takes to recover from injury. So it's all just putting lipstick on a pig in my opinion. So I have given up caring about "mimicking reality" in-game a long time ago because none of these rules are trying to do that. These are rules that exist merely to play a game. That's it. Not to truly
simulate anything, but just to give us rules to play our little dice game
upon which we flavor that dice game with what these rules represent in some sort of fictional story.
But that's just me. I merely handwave all of the disparities and accept it all merely as the tenuous connection between game and story.
Now if you hoping for the latter part of my original statement-- that you are just wanting the PCs to start each day not at full-strength because of
game concerns... that is also understandable. If you are having a hard time challenging your players because when they are at full strength and capability they have too much "stuff" they can do to run roughshod over your built encounters... forcing them to start more encounters with less "stuff" is one of the easier ways to make encounter-building easier on you. For a lot of people they just stop allowing characters to take Rests as often... rather they force their players onto the next part of the adventure via time pressure or the lack of safe spaces to Rest. Thus the PCs start many of their encounters at less than full strength. Cutting down on the number of hit dice they have is certainly one way to do that... but I will be honest, I myself do not expect that this
particular house rule will properly accomplish this hope, for two reasons. One, players are notoriously risk-averse. As
@Horwath said above... if they can't get back to full-strength overnight... they'll just sit around taking as long as they need until they are at full. At which point you as the DM have to institute time pressures and lack of safe space to Rest anyway just to force them to get back to the adventure rather than wait until they are healed. And two... hit dice are actually some of the least important mechanics to curtail if one hopes to reduce PC power during a gameday. Because they only affect things after the encounter is done. If you don't inhibit any PCs active abilities, then they will still be fighting in every encounter with their entire suite of features, and thus making your encounter design still difficult for you. You'd actually be better off not slowing down HD recovery, but HIT POINT recovery instead. Have the PCs lose a point of their "Hit Point Maximum" (like what happens when fighting a Wraith for example) from every single hit they take. That would actually reduce the PCs strength in subsequent encounters and allow you to challenge them more.
So the long and the short of it is that your reasons for doing it are valid for whatever you think you want... but I just don't know if this particular house rule you've come up with is the best way to do it. My instinctive impression is just that it isn't going to accomplish what you want it to. But hey! This is why we playtest! Try it out at your table and see if it does indeed give you what you want. And if it doesn't? Go back to the normal rules. I do that all the time myself... try out house rules to see if I feel better about the game when using them. And if I do, I'll keep them (like the Variant Ability Score for Skills house rule) and if they don't really improve anything in the game I drop them (like rolling 2d10 instead of 1d20 for skill checks.)
Good luck!