The inefficiency issues are part of what makes it interesting. Yes sometimes you will will not roll enough and other times you will roll too well. But combat is random. Sometimes your parry attemptes aren't as good as you thought they were and other times you spend a lot of effort dodging an attack that wasn't much of a threat to begin with. Of course, I created the optional modules for those who want to lower the randomness to either use static HD or static damage rolls.I feel like this would make weak enemies much more powerful than they would otherwise be. If a goblin hits for 1d4, then it's not generally worth spending a Hit Die to block that, where you might be more willing to spend a d10 to block an orc or ogre. Personally, if I was playing this game, I would use the rule for average values - nothing worse than spending a d10 to stop 5 damage and only getting a 1, except possibly spending a d10 to stop 5 damage and rolling a 10. Massive inefficiency ensues.
That is my suggestion for "stamina". If you don't want to roll HP for each attack, you roll it once and it lasts until you take another short rest.Why not just roll your Hit Dice together after each short rest, and make them act like temporary Hit Points? If you're level 8, then you might have 20 Hit Points and 8d10 Hit Dice, so just say that you have 20 Hit Points and 44 temporary Hit Points?
Also, you probably don't want to increase Hit Points by (1 + Con modifier) at each level. Aside from killing some people outright when they reach level 6, it places way too much dependence on the Con score.
I am honestly fine if 6 Con people are unable to be adventurers. 5e is already so harsh on anyone who has a Con score of 10 or less anyway that I really don't see this as an issue. Most PCs will have a 12-16 Con anyway, as point buy (and die roll) easily allows for a tertiary Con score of at least 13. So I have no problem that those who are sickly and frail (Con 8 or less) are never adventurers.