Hit Dice were pretty clearly intended to do more than they did in 5E, but at this point, I suspect we're stuck with them being a weird little vestigial subsystem. I'm not sure you should be able to spend them to get more offensive punch though.
I mean, it's a real pity that they're so vestigial. A number of other games have similar mechanics which work well (System Strain in Worlds Without Number, for example, or yes, Healing Surges in 4E), but I don't anticipate 1D&D doing anything interesting with them at all beyond possibly letting you spend them outside short rests for healing more. I notice the only major real change involving them so far is that where in 5E and the very first packet, you regained 1/2 HD on a Long Rest, in the two more recent packets, you regain all HD on a Long Rest.
Personally I'd really like to see most levelled healing spells give you an opportunity to spend 1 or more HD in addition to the healing they do, at a minimum.
I am not a big fan. My biggest issue with it, is that making hit dice a pool of dice you can spend for more power, makes have a dedicated healer character necessary. One of the best things about 5E is a party not having to worry about who has to play the cleric.
This is certainly a risk. Right now, unless you have a Life Cleric (or wacky house rules), it's likely the majority of healing your party does is via Hit Dice and Long Rests. But if people could spend them to increase power, it's likely they'd blow through them at an incredible rate. Life Clerics are thankfully smacked with the nerf bat in 1D&D, but nothing will convince certain players that relying on others to "heal them up" isn't fine and dandy (MMORPGs and CRPGs have really not helped with this lol), so I think it's something that needs to be addressed mechanically - and that means avoiding translating Hit Dice into really anything but defense/healing abilities.