The problem with that is that you'd be double-dipping on levels.
Let us, for the sake of argument, say that we'll go with the progression you suggest and extrapolate it: 1 HD per 3 levels, rounded up. Let's then look at the middle of each interval (2, 5, 8, 11), and let's further assume that we're talking about a fighter with Con 14 and using the 2014 standard of max hp at level 1, and half max +1 at each subsequent level.
Level 2: 20 hp, level 1 cure heals for 1d10 (average 5.5). 27% of max.
Level 5: 38 hp, level 3 cure heals for 6d10 (average 33). 87% of max.
Level 8: 56 hp, level 4 cure heals for 12d10 (average 66). 118% of max.
Level 11: 74 hp, level 6 cure heals for... well, kind of irrelevant because at that point you have heal.
I think either I communicated my approach wrong or you misunderstood.
At 1st, 2nd, and 3rd level, a cure light wounds does 1d10 on a fighter and 1d6 on a wizard
At 4th, 5th, and 6th level, a cure light wounds does 2d10 on a fighter and 2d6 on a wizard
At 7th, 8th, 9th level, a cure light wounds does 3d10 on a fighter and 3d6 on a wizard
A cure serious wounds (the next one up I am assuming I don't own 5e)
At 1st, 2nd, and 3rd level, a cure light wounds does 2d10 on a fighter and 2d6 on a wizard
At 4th, 5th, and 6th level, a cure light wounds does 4d10 on a fighter and 4d6 on a wizard
At 7th, 8th, 9th level, a cure light wounds does 6d10 on a fighter and 6d6 on a wizard
A cure critical wounds (the next one up I am assuming I don't own 5e)
At 1st, 2nd, and 3rd level, a cure light wounds does 3d10 on a fighter and 3d6 on a wizard
At 4th, 5th, and 6th level, a cure light wounds does 6d10 on a fighter and 6d6 on a wizard
At 7th, 8th, 9th level, a cure light wounds does 9d10 on a fighter and 9d6 on a wizard
You can massage the numbers to get different results, but they'll always have this sort of shape. The clever thing with 4e's healing surges was that they were sort of independent of level – they were always 25% of max hp, and you didn't get more of them as you leveled up. Something like that is basically the only way you can have both meaningful attrition and healing that's relevant in combat.
The healing amount though is just like a surge. I use dice because it's more D&D like but that is the idea. The difference is you get unlimited surges but they can only be activated by magic. But for those of you liking non-magical healing, you could have one rest do your hit dice in healing. You could have a long rest do half your level rounded up in HD of healing.
3e gave clerics the ability to swap out prepared spells for cure spells, which was meant to have this effect. The result was that clerics loaded up on buffs and wreaked (wroke?) havoc.
I've wondered if healing as a subsystem should be separate from spells. I haven't decided yet.