Heroes 5.5 WarGame Edition - Tactical Edition.

I disagree. Let the Knight drop defense and add machines. This would give the class a focus on an aggressive style. Let the Temple hero drop machines and add defense.

There was such a proposal—to drop defense. But several players (myself included) had very serious arguments against it: a knight without defense is not a knight. Even if it is an attacking class. In other words, this point is fundamental (a hard contradiction).

Defense is also a poor fit for the Temple hero. The Temple hero is a line of monks. If the knight also has defense, then we would have 3 classes with defense, whereas defense should not be a racial skill (it is racial for dwarves; for humans, it is leadership).

So are they Temple heroes or fanatics? If the focus is specifically on zeal in a religious sense, rather than the fact that they are clergy, then it's better to call them fanatics, as suggested above. If the essence is in the profession, not the outlook, then the education is appropriate.

Definitely fanatics. But I don't understand why "either/or".
The word fanatic doesn't explain the magical orientation and the extremely high SP (40%). And a class with such a name would be more suited to face-value leadership (20%), not wizardry. Which isn't great because of the similarity to the paladin's face (Lead+Light+Dark vs Lead+Light+Def).

And does he definitely need wizardry? The synergy with dark and light is mediocre anyway.

All classes with 40% spellcasting have wizardry (the same applies to attack, defense, knowledge—and their corresponding skills).
Wizardry is useful for Light and Dark because in WGE, the main strength of this skill lies in the abilities (keen mind; the mana branch, which is critical for low knowledge; the spellcasting branch with a total boost up to +6, which is important for short-term casts).
And let's not forget that mass spells are an expensive pleasure; mana is scarce, and humans with 10% knowledge certainly won't have enough mana to use them constantly, whereas when target-specific casts come into play, the skill itself will matter.

That's the point; having more than two schools of magic is pointless. It gives almost no tactical variability or direct bonuses.

Under the bonuses from those abilities, it turns out it's not pointless.
Approximate version: Light+Dark+a third school.
Mass buffs: stone skin, possibly evasion; mass debuffs: weakness, curse, slow, possibly distraction; and then kill with a phoenix/elementals/chaos, etc.

And if the "magical omniscience" skill remains in the game, the archmage will have negative synergy with it... Which looks completely strange.

Doubtful. Yes, part of the schools' effect disappears—the ability to learn new spells. However, the main strength of schools is the enhancement of spells, which combines with passive bonuses up to +8/+8 from education abilities.
These +8/+8 are equal in power to the original +16/+16 (since each characteristic is twice as strong in terms of balance). I doubt an archmage with such a bonus will suffer much from extra schools, especially since he doesn't even need to max them out—just taking them is enough.

Therefore, my thought is to give the class a bonus to compensate for these shortcomings compared to other classes of the faction.

I don't see any shortcomings here, since generally, a build via education+wizardry+two schools of magic works for many other classes; the archmage is not unique here.

Added after 9 minutes

"Heretic"? "Centurion" (a leader of a "legion", that is)?

Heretic fits Heroes 3, where demonic heroes were humans who betrayed their faith. But a native demon clearly cannot be a heretic, unless he turns toward the Light :)
Centurion/Legate? Sounds Roman. I can't say yet if it's good or not.
But the faction is indeed called "Infernal Legions".

Added after 2 minutes

What about Agrael's initiative?
Gating is magic, if you think about it. And in my opinion, it's no longer for meat (you have to build him up very well to crush with meat), but for tactical opportunities. Blocking enemy shooters, keeping your main stack out of the line of fire, removing retaliation... I think for a hero with a magic lean, this arrangement works.

Agrael is a campaign hero with an uninteresting, duplicating special ability (similar to analogues); you can do whatever you want with him. A different matter is a standard hero with a unique special ability.
Too many specialists on initiative. Wingaël (starting turn) will definitely remain and perhaps there will be one more for initiative (penalty). But no more. Initiative bonuses already conventionally exist for Vinrael (who has the Haste specialization).

Regarding gating, Nimus crushes specifically with meat. Since his specialization does not qualitatively enhance gating, only quantitatively. Like a combatant.

By the way, in a neighboring thread (MMH5.5), gating is implemented that summons a smaller number of creatures, but faster.
The summoner spends 75% of the turn (instead of 50%). The summoned unit starts with 75% of the accumulated turn (instead of 0), thus beginning to act after 1+0.25 = 1.25 turns from the moment of summoning (instead of after 2 turns). The amount of meat is reduced to 15-25%.
I don't really like the numbers (about 75%), but what do you think of the idea itself—is it worth trying to port to WGE? Or is the old gating with a large amount of meat, but delayed, better?