How does using such a "skill tree calculator" (Yes, in this case, let's call it that, because people who have played more games will remember all the "non-standard" skill trees and know where to get what they need and where not) differ from using a guard calculator, a damage calculator, or Phizmig?

In fact, there is one difference. What is described in Phizmig – the algorithms for calculating damage and generating guards – is part of the game's mechanics. These are things that the developers put into the game themselves, either intentionally or by mistake.
But skill trees are no longer an element of the mechanics. In fact, the developers did not put any skill trees into the game. Skills are given to heroes randomly; the developers simply decided to make it so that by loading the game and playing differently, you could not get the desired skills when leveling up. As a result, there turned out to be not very many different possible sequences for obtaining skills, and because of this, it is possible to predict which sequence has appeared. And this possibility is essentially the use of a vulnerability of the random number generator to predict the next numbers in the sequence.

It is quite possible that the developers (if they still existed and maintained contact with the players) would share information about the formulas for calculating damage or guards in the game. But it is difficult to imagine that they would reveal the formula for generating a random number for a hero's skill to appear.

Templates are a more interesting analogy; they really played a role as a regulator of randomness. But their idea was greatly developed, and it is quite possible that they were intended to be something more than just a simple regulator of randomness, but the developers were unable to complete (or continue) this, so the players had to figure it out on their own.

There is also something like predicting obstacles (and the presence of a stacked item) on the battlefield based on the coordinates of the cell and the terrain – this is already a direct analogy to predicting a skill tree; it also uses a vulnerability of the random number generator.

P. S. All of the above is just a perspective from a person who has delved into Heroes and understands where these features of the game come from technically; perhaps someone will find this interesting. I perfectly understand that sometimes it is more productive to ignore what the developers intended and I do not claim that the above should be taken into account when determining what will be better for the players.