AI cheating on single-player maps.

Yeah, sometimes there are ridiculous bugs where the AI flies or walks on water. I've noticed that this happens more often on random or custom maps, and less often on standard maps. I had an interesting experience on the "Heroes of Sword, but not Magic" map, where magic was completely forbidden, and the enemy AI somehow used a magic arrow in one battle. Unfortunately, I don't have that save file anymore. In general, if the computer didn't cheat so much, it would be even easier to win. After all, the AI is a computer, and the restrictions on the maps are more for the player than for the computer. So, if it's more advantageous for the computer to do something, it will do it, even if it goes against the rules. The computer rarely digs for the Grail, if I'm not mistaken, it digs it up if it has explored and discovered enough of the game world and visited several obelisks. Since the computer mainly acts linearly according to a program, if the goal is to destroy opponents, the computer will do just that if it has the opportunity. But yes, the computer compares the fragment of the map it has opened with the fragments of the obelisk map, so it only needs a few fragments of the puzzle map. But the thing is, it "knows" where the Grail is buried, but it doesn't immediately run to dig it up; instead, it continues to carry out its programmed actions. For fun and to test, you can try a map where you need to find the Grail to win, and then see how the computer will act. But even there, the computer doesn't seem to be particularly eager to dig up the Grail. Therefore, even from a technical point of view, it's difficult to talk about the computer having an advantage over the player in finding the Grail.

Added after 5 hours 6 minutes

The AI looks for the Grail out of boredom. If it is forced to defend, it will look for resources, troops, and upgrades...

It's not quite like that; in general, all the AI's actions are described by a simple algorithm. This is how the computer explores "terra incognita," gathers resources, builds, and so on. In general, if it has the opportunity to visit several objects, the computer first chooses the most important one, what it needs, and in a certain sequence, the AI collects everything. If it has the opportunity to visit or collect something, the computer will usually do so. Priorities are largely determined by the assigned behavior. The AI can be: a warrior, a builder, or an explorer. All AIs will do roughly the same things, but with different priorities. For example, a warrior will first gather resources, build troops, and look for upgrades, and attack neutrals. An explorer is more likely to run around with a group of weak heroes, explore the map, visit everything, including looking for the Grail. That's roughly how it works. The computer's action program looks like a simple sequence of actions, so the computer acts quite linearly and rarely surprises.