I think there's a simple issue to sum up the problem - implementation details should be hidden from the user. But AI systems that rely primarily on excessive cheating become a game where knowing their implementation is essentially 100% of the strategy because overcoming such an unfair gulf is basically impossible without breaking the AI. This lends itself to awkward strategies that should not work, but do. And it simultaneously also makes many aspects of the game basically red herrings in that they simply don't work. And that's seriously (!!!) a shame because the designers in Civ have done a great job of creating lots of really interesting systems, but these systems have gone far beyond the AI's capability, which is just compensated for by extreme cheating.

It can also lead to the appearance of buggy behavior if you don't understand the implementation. For instance I think Firaxis did a much better job of the AI in XCom (in that it's enjoyable to play against at least), but there too you kind of need to understand the implementation or you might wonder about why, for instance, enemies seem to be magically teleporting around the map. The answer, of course, is that they indeed are.

Of course creating a non-cheating AI that's competitive against skilled humans is extremely non-trivial. But if you at least work on improving the AI you can tune down the amount of cheating necessary to create a challenge, which means the experience can feel much closer to 'organic.'

Because it becomes very not fun to play with the AI when the unfair boosts make it very difficult to win without just cheesing it in ways that exploit its predictable stupidity.

It matters because it becomes very quickly obvious. And the moment that it becomes obvious that you are not playing against an opponent that tries to use cunning and strategy, but merely an equation - a puzzle - which you have to solve, the entire house of cards comes down. Puzzle games can be fun, but the very people in this thread are arguing that CIV games are not about being a puzzle, but about roleplaying a civilization. There's a puzzle aspect of course, but a blatantly cheating AI is just a puzzle and nothing else.

It matters for the same reason that people don't want to play against a human-like AI, because that AI would be a double crossing bastard who tries to win at all costs. They want to play against a roleplaying AI who tries to appear like Ghenghis Khan or Tutanchamon. But if I see my sworn enemy Attila the Hun spawn in an endless stream of battering rams out of thin air, and yet is so utterly incompetent and dumb that it cannot navigate a simple pass between mountains, that's a worse transgression against my suspension of disbelief than if he was just plain bad at the game.