I go back and forth a bit on this, myself.
I do think there should be ways to lose HD without, y'know, using them at rests. It feels to me that this was a mechanic that was changed late in 5E testing, and never "fully-integrated" into 5E's rules. I think in early Next testing, weren't they more like Healing Surges? I suspect they got removed in some internal or NDA'd test, then added back in to the final design.
They just feel a bit disconnected.
I also agree re: healing spells, though I dunno if them burning HD is the right way to go, because I felt like in 4E, that was an ineffective mechanic. I suppose thought that it does a good job of keeping things balanced, and it makes healing spells less central because you're kind of borrowing from your future self (and whilst you probably get a bonus, you also lose out on a spell slot). I do think healing stuff which is based on d8s or multiples of 2d4 should pretty much universally actually be rolling appropriate number of the class HD in place of that (and some d10 healing stuff too). If a spell heals for 2d8+WIS, it should probably actually heal for 2HD+WIS.
With 4E I think the biggest issue was that you got too many, like kinda way too many. CON mod shouldn't have been involved (it's double-dipping because you're also going off a fixed HP value modified by CON), and the numbers should have been a bit lower (like maybe 4-8 instead of 6-10 base).
The problem in using them for anything in 5E, though, is that they're linear, instead of a fixed number. In 5E, you get 1/level. If instead of that, it maybe started that way, then capped out at a certain point, they could be used for environmental damage or the like. But capping them out doesn't work well with burning them for healing spells, balance-wise.
I think the main integration they could have for a 5.5E or a close-to-5E 6E would be to use class HDs in healing spells wherever possible. It's fine to have some spells not do that, especially wacky ones like Healing Spirit, but I think the "basic" healing array should all be based on multiples of your actual hit point die (as should healing potions).