This system captures the idea embodied in the surge mechanic (the out of combat "breather") without using surges. It circumvents the martial healing problem by simply making non-magical healing more effective than it was in older editions.
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Thoughts?
There is a basic problem with this type of system (which is similar to the 4E issues). The Breather doesn't just heal stun type damage, it also heals real damage (since they are combined in this type of system).
If you have a 6th level Fighter in the front taking damage every single encounter and you have no magic in the group, that Fighter will get beat up over and over again until he goes unconscious (as an example).
That Fighter has a CON bonus of 3 and the first aid PC has a WIS bonus of 3, so he has 6.5 * 6 +30 = 69 hit points. He goes unconscious, so he wakes up after the encounter with a first aid D6+3 = 6.5 and a breather 5+3+6 = 14 or about 21 total hit points. He's about 70% damaged.
The 6th level Mage with the same CON has 59 hit points. The Mage also goes unconscious, but the Mage gains the same 21 hit points back. He's about 65% damaged. Since he is a Mage, he hides in the back shooting his magic and never takes any more damage the rest of the entire day. Every encounter, he takes a Breather afterwards for 14. In 3 more encounters, he's fully healed, just by taking Breathers.
Without Breathers, this system would work (somewhat). The PC would be able to heal up with First Aid (once) and would never heal up all of the way (assuming one could not First Aid for more points in an encounter than the total of damage taken in that encounter).
The Fighter in this example presumably takes at least a third of his hit points every encounter and ends up unconscious every encounter and back up to 21 hit points after each encounter. After 4 plus encounters, he's at the end of the day pretty darn close to 21 hit points (give or take a handful or so).
So, this system tends to break down in multiple ways.
Original D&D hit points had no First Aid (which also has some minor problems) or Breathers. Hence, it did not have these types of issues. It was magic or nothing, so magic (or long rest) was required.
4E has Breathers out the ying yang, so it doesn't have the problem of a group without magic having PCs 2/3rds damaged going into encounters. As long as the group has healing surges, every PC can be at max or near max hit points at the start of every encounter.
In the system I proposed above, hit points always goes back up to full, regardless of how much damage one takes. The Mage does not have an advantage over the Fighter except that the Fighter might eventually get real low on wound points and would want to rest. But, wound points stay lowered until really healed one way or another. A Heal (i.e. first aid) check in combat or after combat doesn't heal any wound points (although it could be added in as part of an extended rest to slightly increase how many wound points get healed during an extended rest, 1 wound point without first aid, 2 wound points with first aid makes sense, wounds shouldn't heal too quickly).
One thing that WotC didn't do a real good job of in 4E is analyzing the math really carefully. I hope they do that in 5E. Your example illustrates how a system can appear fine on the surface, but can have underlying issues under certain circumstances (in your example, a party that has no magic) because a more typical situation (a party with magic) would obscure some hidden issues.
That's also why a LOT of playtesting is required to shake out more of the more common design flaws. There are just too many variables in any RPG to figure out most of the potential issues just by reading the rules. Some flaws can be discovered that way, but not most of them.
Note: Even my system above has at least one minor flaw. If wound points = CON and PCs heal 1 wound point per extended rest (2 with first aid), than a PC with 10 CON having taken 9 damage needs 9 days to recover whereas a PC with 18 CON having taken 17 damage needs 17 days (or about twice as long) to recover. Course, the 18 CON PC took 17 wound points. The 10 CON PC doing this would have died at 10 wound points, so pros and cons. The point is, though, most any system has at least some minor issues with it. No system is really perfect.