With the exception of one life cleric with mass cures, almost all in-combat healing I've seen has been just to stand someone fallen and very little proactive healing.
True. The Heal-from-Zero rule strongly encourages standing the fallen over healing proactively.
(So y'could just drop that one.)
But that doesn't address what I believe is the root cause that it doesn't heal enough for the action spent.
OK, yeah, also a factor...
PROPOSED HOUSE RULE:
When you are the recipient of a magical healing spell of 1st level or higher or from another magical source like a healng potion, you may also spend HD up to the number of dice from the healing.
Sounds fair. HD are a very limited resource, though, so it might not see that much use, and it adds complexity to the decision to heal, since whether the target has HD remaining and wants to use them becomes a factor.
I was worried about martial types using up all HD and pushing for a 15 minute adventuring day, but then it occurs that they'd be spending about the same HD during short rests anyhow.
That's a question of resource management that your change shouldn't much impact. Ultimately, HD are a limited resource, and the slowest-recovering resource, so they can't do a lot of heavy lifting.
Increased magical healing will likely have a response of an increase in challenge,
I wouldn't think so. Proactive healing can be less efficient, so it shouldn't make the party wildly more potent than reactive healing of downed allies.
As has been noted, burning HD in combat will reduce the impetus for short rests, which could impact characters with significant short-rest recharge resources, like the Fighter and Warlock.
more dependence on a magical healer unless there is also a way to do non-magical healing.
Dependence on a magical healer is a D&D tradition, anyway. Diluting that with a non-magical alternative is, IMHO, a separate issue from the use of healing spells in combat...
PROPOSED HOUSE RULE:
A new action, Rally, is allowed for all characters. When you Rally you may spend HD up to your proficiency bonus. Once you do so, you may not do so again until you complete a short or long rest.
Sounds fine, reduces the need for magical /proactive/ healing (so, again, you'll have less of it), and since you can't rally yourself from 0, makes conserving magical healing for dropped allies a good idea.