I just find the entire concept of the Warlord annoying. Not only do I not want to play a Warlord, I don't want to play in the same group as a Warlord. Maybe that is a bit selfish, but that's the way I feel.
That is selfish, yes. You want to play the character you want, but do not want to extend that same courtesy to others. Not only that, but you want to deny everyone, not just anyone you might game with, the option of playing a character they might want to.
As the OP asked, why? What makes you so much more important than everyone else who's ever played the game?
I do understand the concept of the Warlord, I understand the RP, and I understand that people like the idea of a support character with no magic. What I don't understand is why people want a non-magic support character.
One very obvious and practical reason is for campaigns where one or more of the existing, magical support classes are inappropriate. More personally, it's a good concept, one quite common in genre, and fun to play.
More pedantically, for the same reason you might play a Champion, Battlemaster, or Assassin who doesn't use magic. Because the character concept doesn't include magic.
The same thing that's wrong with psionic or martial abilities: nothing. They're all legitimate.
If you want to play a character who uses magic, you have /many/ options, over 30 in the 5e PH, alone. If your concept doesn't call for magic, your options are much more limited, only 5 sub-classes in the PH. If you wanted a psion, the PH left you out in the cold.
Equally valid concepts, profoundly unequal options available to model them.
My hope for posting this is to give some insight into why someone (me) would not like the concept of the Warlord and objects to it even being an option. It may not be a good reason, but it is what it is.
It's easy to see not liking the concept - you've failed to given any reason for it not being an option, at all - beyond simple selfishness, which, yes, is a bad reason.
Well I don't want to be bogged down in semantics. I'm just trying to give some clarity to the anti-Warlord position.
Oddly, that's not what the OP actually asked about:
The question is this... I don't get the idea of being opposed to something that many people clearly want. People limit things like races and magic items and things like that. So why can't people just disallow the classes? Think martial healing is terrible and shouldn't be a thing? Don't use it. It's literally that simple. It seems childish to not want something but then force your opinion and views on other groups who have completely different expectations and playstyles. Can someone clue me into this kind of mentality? Is there something I'm just overlooking? Does the game actually force you to play these classes, was it a rule I overlooked or something?
Now, maybe that wasn't a very fair question...
And I've offered some ideas on what would be more acceptable to me. As expected they aren't what Warlord players really want. And that's the problem. Neither side can come to a good compromise. So not either sides fault, just that the two sides are still too far apart to meet in the middle somewhere.
The compromise is to base the Warlord on what people who would actually use it want, and for those who dislike the very concept to simply not use it. That compromise works nicely for everything else in 5e, and could work for other things it hasn't tackled yet.
See, I just don't understand this. You are using the power of your spirit to push your muscles to the limit. Your spirit gives you the mental fortitude to continue on despite fatigue or fear, magical or otherwise. Why can't your spirit force the magic that is in your body, and in everything else all around you, to bend to your will and allow you to do the impossible?
If you lift an unliftable gate, does it matter that it is your muscles lifting it or the magic that resides in your muscles?
When a Barbarian goes into a rage, is it all just psychology, muscles and adrenaline? Why not some magic to allow him to go far beyond what is humanly possible?
Stepping back, I guess we just want different things. I need an explanation for everything. If that explanation is magic, that's fine.
As you posit it, though, 'magic' isn't an explanation at all, if it's the only explanation /for everything/. It's also not how D&D has ever treated magic. D&D draws this mechanical line. Things magical reside on one side of it, subject to anti-magic fields and, perhaps, to being Dispelled (and in 5e, countered). Other things might be supernatural, but not technically magical - psionics has sometimes fallen into that category. The psionic could use TK or teleportation or whatever other clearly supernatural power, but could even do so within the rare anti-magic field. 3.5, for instance, drew particularly clear lines with the (SU) and (EX) designators, for another instance.
Oh, I also dislike the whole "fully healed after a good night's rest". There are some wounds that can't be fixed by sleeping it off. I suppose if your characters just never take those wounds and it is always, "Well I got banged up a bit (down to 1 hp), but I'll be fine!" it works. But that really seems to limit your descriptions of damage.
It does, yes (though not a whole lot: any wound that might be ignored in the heat of combat or so much as stabilized after an hour's or a night's rest is on the table). In that sense, hps are a lot more like a model for 'plot armor' than for actual injury. The DMG has some modules that let you change natural healing around - doing so, plus maybe house-ruling in some lasting wound penalties or the like - could expand the kinds of wounds that hp loss could plausibly model.