AbdulAlhazred
Legend
According to KD it is historically typical that characters lose half their hit points every encounter.
I think you are not looking closely at this enough. This is an attempt to keep adventurers going without having to rely so much on clerical healing. Though it still can be important to have and viable to have healers because wounds can only be healed by clerical magic or an extended rest.
It is also a way of saying that some hit points lost are the result of fatigue and not just purely physical damage (this doesn't jive for some and that's ok). So there must be a way of getting those hit points back quickly by taking a short rest.
To me it makes sense to have a classes that can heal not so much healing classes. Clerics should not be pigeon-holed into being just healers.
The real question would be whether or not non-magical healing would even be worth bothering with. I get the feeling it wouldn't really. At best you're restoring a resource which will restore anyway. It might be useful to be able to do that mid-battle, but TBH that sort of non-magical healing is the most 'gamist' of all (really, try to do some 1st Aid in six seconds, really). I know it has been long accepted, but it is pretty hokey really.
So, the problem is pretty clear, you're going to HAVE to have magical healing to go on for long. Not perhaps to the degree that it was required in previous editions, but there's little chance your warlord is replacing a cleric, you'll need one, or at the very least a supply of potions.
As for the 'squishy' thing, I don't really see any reason why a wizard should be that much squishier than a fighter really. They are both adventurers, not desk pilots. The fighter may be somewhat tougher but they're all heroes and they'll all tough it out. Squishy magic user is trope that was born out of a failed attempt by AD&D (or maybe we should blame Holmes Basic) to balance magic users against non-casters. It really has little justification and in a game where the martial characters can do cool things is mostly unneeded. In fact I think the game would benefit from a bit easier bleed-over between the two concepts. A wizard should at least have the option to be tough enough to pick up a sword and a clever fighter could easily learn a bit of magic too for that matter.