@Flamestrike I'm in this thread to help the OP theorycraft a system to bring 4E Healing Surges into 5E. We obviously disagree, and there's zero to be gained from going around in circles and derailing this thread. So unless you have something constructive to contribute, I'm going to stop responding to you.
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Back to the point...
My math is running under the assumption that the average adventurer has a d8 Hit Die and 14 Con. I'm also looking at Cure Wounds as my healing benchmark. By converting it to spend Hit Die instead of rolling a d8, the following changes occur:
Cure Wounds becomes less effective for wizards and sorcerers, and more effective for martial classes. Since martial classes are more likely to take damage, I'm okay with this. It makes combat healing slightly more powerful for the characters that it would actually matter on.
On the other hand, on a character with no hit dice remaining, Cure Wounds becomes significantly less effective (only restoring the spellcasting modifier value). That makes the timing of healing magic more important, and complicates the decision as to when to use healing magic and when to push through and try to heal naturally later.
At first level, assuming the above averages, a cleric can cast Cure Wounds twice, and the average character has two additional hit die, so that breaks even. But thinking about it, the Con bonus hit die should probably be d8, not d6.
Overall, magical healing becomes more useful inside combat, and less useful outside combat (where it now directly competes with the resting recovery mechanic).
Related, I like the idea of giving life clerics something similar to the Bard's Song of Rest. Even a single bonus point keeps some of that Life Cleric flavour.
In terms of number of hit die, there's a number of different methods we could use, depending on what feel we're going for. Giving them most or all of their total hit die up front makes early characters more survivable, but also front loads said surviability, and makes it harder for more durable classes to be represented in the mechanic. My current idea ( class hit die + [Con mod]d8 ) gives a smaller pool up front, but keeps the different between healing a wizard and healing a barbarian intact without massively overpowering level 1 martial dips, which I personally prefer.