Bacon Bits
Legend
I've yet to see any kind of reason why we should not have a Psion, Gish, and Warlord.
Psions are difficult. Most of the time we see that people say they should have a system separate from magic, but the balance of the magic system (such as it is) is actually pretty difficult. It's going to be very difficult to retain the existing magic system and add another system on top of it. The proof here is that there's barely a mechanical system for martial classes at all, and it's just outclassed by magic. Magic has a bunch of artificial drawbacks (concentration, spell slots, costly components, spell level requirements) to try to keep it in line with martials, and it doesn't entirely work successfully. The martial systems (extra attack, sneak attack, feats, class-specific damage bonuses) are simply designed to not scale as well as magic. Adding a third system to that is going to be quite difficult, because you probably can't make it more restrictive than magic or more freeform than martials, and threading the needle between the two risks eclipsing one or both systems. Which is exactly what happened with 1e AD&D psionics (they were either irrelevant or busted) and the solution for 3e was that psionics were just magic but with more math. Further, you have the problem that Basic D&D, AD&D, Dark Sun, 3e, and 4e all have completely incompatible ideas of what psionics are or how they work, so there is nothing to draw from for a psionics class that will satisfy everyone. WotC will have to pick something and go with it, and the current development teams are built around not doing that. They're built around accepting nearly infinite feedback before publishing. That design model is incompatible with such a diverse mechanical history as D&D's psionics. It can't be done by WotC until they set that aside.
Warlords are probably not workable at all. Or, rather, it you made one it would probably work more like Miniatures Handbook's Marshall from 3e. Warlord works well in a high tactical system like 4e, and in a system where the Basic attack is a relatively fixed constant independent of class. It means the class has a lot to do, and the tight balance of 4e means you have an interesting choice each round. In 5e, movement, positioning, and tactics are markedly less potent. Whole classes exclusively rely on the standard attack roll, too. This means not only is it going to be less interesting or beneficial to grant bonus movement, it means that granting an additional attack allows you to deal damage is even better. In other words, your action makes you deal as or more much damage as the Rogue or Paladin does on their normal turn, only you also get to do everything else that a Warlord can which presumably has to be more interesting than just clone Sneak Attack or clone Smite. That makes the class incredibly difficult to balance, since it also has to function in parties that don't have one of those two classes. The obvious solution to have the Warlord's Commander's Strike use it's own fixed damage and have no benefits from the PC would be balanced... and also so incredibly boring and disappointing that nobody would be happy.
Gish could exist, but is unlikely to do so in a significantly different form than what currently exists. Ranger But Arcane, Paladin But Arcane, Bard But Vanilla, Cleric But Arcane, Bladesinger But Fighter, Bladelock But Different, Artificer But Different, Echo Knight But Better, or Eldritch Knight But Better are not really character class design concepts. That's really all I see when people ask for a gish: Explanations for why X isn't a gish. Well, it's rather tough to define something by what it isn't. The trouble is that "gish" is a different concept not just in every setting, but for every player. There is no such thing as a generic universal gish that will satisfy every player. Even if they literally publish a class named "Gish" and people will still say, "We don't really have a good Gish." It's a broader concept than Ranger is, and just like ranger the Gish always has unique trappings in every portrayal. It's one of the reasons Rangers kind of fail in 5e; they're too complex to lack a single vision because the concept is too broad now. Simply put, the game already supplies multiple classes and subclasses to support a mixed martial-spellcaster. It's overwhelmingly the most common example of a D&D character type in 5e to the extent that nearly every party will consist of characters capable of using weapons and spells in tandem.