AD&D you did not heal over night though or get hit dice based healing.
Making clerical healing that much more critical, yes.
Otherwise see previous comment about the expectations of the amount of combat or otherwise dangerous stuff.
Saw it, but I see no reason to throw away all my experiences playing AD&D for 15 years, and the similar experiences of everyone I've ever talked to in person about that aspect of the game, based on the unverifiable anecdotes of one anonymous forumite. Nor do I see 'simply' adjusting combats/day as a solution, as doing so impacts both class and encounter balance. You're looking at major re-tooling of the game, if you want to greatly reduce available healing.
In regards to magical healing I am finding it hard to work out how much healing the warlord should have.
You could start by figuring out the magical healing that each of the primary magical-support classes (Cleric, Bard, Druid - maybe Paladin) can provide. There's more than just Cure Wounds to take into account, so it might not be quite as easy as it sounds. Then take an average or guesstimate at what level of non-magical hp restoration it would take to keep a party going to a comparable degree...
Sounds more like a reverse vampire to me.
Heh, or an empathic healer...
I think a simple "ally can spend a hit die, they can only benefit from this once per short rest" would be enough for the base warlord. Just enough to get someone off the ground, with no extra healing.
Some extra healing somewhere is needed, too, to see a party without other support through a standard 'day.'
EDIT: Second question. So what player, in a low magic game, gets cajoled into the warlord?
Whichever one (or two) doesn't feel like playing a beatstick or backstabber. Really, at this point, the selection of non-magical classes is so restricted and DPR-focused, the problem would be keeping the Warlords in the party to a reasonable 1 or 2.
Third question. Wouldn't the feat delivery system work? Or a list of more maneuvers for the Battlemaster?
More battlemaster maneuvers wouldn't matter until higher level (he can still have only 3), and the sub-class (the structure of a fighter sub-class, in general) simply lacks the resource base and versatility to fill primary-support needs of a party. Feats likewise kick in too late (4th) and do too little to cover primary-support. I suppose you could have a whole party of variant-human Battlemasters each with a different hypothetical support feat, and all with a heal-from-zero 'maneuver,' and maybe get by - if nothing else, they'd kill most things very quickly - but that'd hardly be enabling the play style, just a bizarre hypothetical corner case.