Heroes 5.5 WarGame Edition - Tactical Edition.

As I understand it, the Knights are not placed on tier 6 in the original game because they are cavalry.

Knights are knights, and heroes are heroes. You can't have super-knights in your army without having regular knights. Because super-knights come from the ranks of regular knights.

It's worth noting that human commanders are also, for the most part, just guys on horseback, but according to the game's balance, they can take down a dragon in one hit if they level up.

A commander is not necessarily a great warrior. In WGE, it is impossible for a hero to take down an entire dragon in one hit, except through leveling up "unstoppable force" or "deadly strike." Without leveling up "heroic attack," even a tier 6, high-level hero cannot take down a dragon in one hit. The combat abilities of a level 1 hero are practically the same as those of a regular knight.

That is, the point here is more about how well a particular type of unit is trained (with a fantasy twist), and the presence of a mount is more of a consequence.

Knights are "very well" trained by default because they are elite professional warriors who can afford the best equipment (foot soldiers in plate armor are also well-trained, but they are slightly weaker and poorer and cannot afford a warhorse). It is unclear what "fantasy twist" means here – is it the mandatory addition of magic, miracle potions, or enchanted armor?
In my opinion, knights should be what they are – knights, without any addition of magic (except for paladins, but they are less combat-oriented).
In the Middle Ages, knights in different countries could vary in the quality of their equipment and training; in general, heavy cavalry is significantly weaker than it is portrayed in WGE for humans. For whom it is assumed that in this kingdom, military affairs are thriving, and swordsmen and knights undergo the highest quality training (otherwise, swordsmen and heavy cavalry would be about twice as weak, no better than current archers and monks). Historically, an archer is often valued more than a simple swordsman.
But in the game, human archers are, on the contrary, mediocre due to the lack of specialization – otherwise, the transition to crossbows would not be justified (specialized English archers would not be inferior to elven archers, due to the greater power of the bows and better protection).

In this whole story, what bothered me more was that such "commandos" are not available in all factions, and in some places they are replaced by non-humanoid creatures, as if they cannot train their "commandos" to the same level.

In WGE, there are many such "commandos" comparable to knights:
8 = priest (dwarves, magic)
8 = bear (dwarves, cavalry)
8 = predator (dark elves, cavalry)
10 = witch (dark elves, magic)
10 = knight (humans, cavalry)
10 = leader (orcs, infantry)
10 = vampire (undead, infantry)