How can the game be made more realistic?

"Diplomacy" and many others are not used by anyone ever.

Are you serious? Diplomacy is a wildly unbalanced skill; it's banned in all communities that have even a slight understanding of the game. Specifically because of the free troop joins. Of course, you can play "without rules," but a game without rules is a completely different story.

1) The vast majority of mass and most frequently used spells belong to Water Magic and Earth Magic. I assume no one will argue with this.

I will. Earth in first place is a must-have and non-negotiable. But Water?
Air would be better. Dimension Door and Fly. Thanks to these two spells alone, Water is ALREADY worse. And then there's Haste, Air View, Chain Lightning. What's in Water? Healing/Remove Curse? Or maybe Clone/Prayer/Teleport? Those might work well in a specific situation, but Air is still "cooler".

Why put skills in the game that nobody needs? There are two options here: either the original concept of Heroes III was different and these skills were indeed intended for active use, or the developers were lazy here as well and failed to bring secondary skills to a common denominator (in terms of usefulness).

Good post. Too bad this topic is in an unpopular section and few people will even notice this post, otherwise they'd tell you :D That this is a brilliant invention, a great concept—if there aren't bad ones, you won't appreciate the good ones, and so on...

Since magic is used 1 time out of 7 cases, it cannot truly and seriously influence the battle.

Please don't make me laugh. It certainly can.

To clarify: if a hero has all seven slots occupied by troops, then for every seven troop moves, the hero can cast a spell only once. Roughly speaking, mages have been weakened seven (!) times compared to knights.

That's a gem. And does it matter that 1 magic cast can directly affect all units? As well as the entire course of the battle.

In the end, a fully leveled Knight will neatly smear the troops of a fully leveled Mage against the wall, especially if the Knight has some school of magic (then he will either nullify the Mage's spells or weaken their effect on his army).

With a Red Ball, definitely. With a Cloak of Denial, it's also possible.
But on the other hand, mages rule on small maps.

What conclusions can be drawn from all this? That I am being forcibly pushed into some frames by artificial means. Am I supposed to sit obediently and replay every map N number of times just so the random number generator finally ends up on my side? No, I won't do that. I will save myself time using Heroes III SaveEditor. Is this cheating? No, this is fixing balance bugs and game flaws.

The fact that you went into the save and changed the mechanics is absolute cheating. It doesn't matter when you did it, on the first day or the last. It doesn't change the essence of the matter. It is cheating. But do it with a bot, as long as your conscience is clear. He won't be offended. He doesn't care.

The part about replaying the map amused me a lot. And the "pushing into frames."
You sat down to play a game; every game has its own physics. And here's the thing: either you accept it as it is, or you make your own game, or you don't play it at all. If you continue to play it, then in any case, the frames and limitations of the game itself apply to you. Constraints, restraints, and words of that nature. Dragons cannot shoot, pikemen cannot fly, and vampires cannot burn with fire breath. No matter how much someone might want that. Likewise, you cannot indefinitely create a Dimension Door and cast magic from a book, say, 10 times at the start of your turn. And many other "cannots" that you "put up with."
Whether you like it or not.

Or am I supposed to put up with bugs too?

You don't have to put up with bugs. But the fact that you didn't get your favorite skill, hero, or magic in the guild is NOT a bug.
That's just how the random rolled. Nothing more. Have you noticed that there is a lot of randomness in the game? And everything that doesn't automatically fall into your favorite subset of spells, heroes, troops, towns, artifacts—everything you don't like is NOT a bug by default. Just because you don't like it (lol). It's simply pushing you as a player out of your comfort zone. You just don't know how/don't want to/don't like to do it otherwise, etc.

So why is any deviation from their opinion immediately equated to cheating here?

Which specific opinion are you talking about now? It's one thing not to use Diplomacy "for balance," and another to crack and violate the save structure to put in what you want, just because you don't know how/don't want to/don't like to play any other way. Is that the same thing to you?

Note that I don't create a hero from the start with the skills I need. That would indeed be cheating. I am only correcting the consequences of the developers' laziness, who themselves (!) force the player to either replay the map or edit the Save Game. Well, or you could put up with it, yes, but why put up with inconveniences that can be eliminated?

"That would indeed be cheating." Oh, really...
I already answered this. Went into the savegame -> changed the skill tree = cheater. Whether it's good or bad from a moral standpoint is completely irrelevant. But a fact remains a fact. Caught red-handed.