Let me talk about this from a subjective perspective instead.
Me and my table of players are old. We enjoy traditionalist games where we focus on roleplaying, the social pillar, intrigue, conspiracies etc. But we also enjoy good old-fashioned combat encounters now and then, so that's a component in our gaming, preferably played out on a grid.
This sounds like you simply do not want what
@loverdrive is describing.
In a typical combat encounter, a player might say "I'm gonna run around that car and shiv the dude hiding behind it". The player then count squares to see if he got enough movement, moves his token, and it's time to roll dice. This is a gaming situation many of us recognize.
Now, if one applied your suggestion about the characters moving like chess queens instead, I would have to reply to the player "sorry, your character Bob the Bruiser can only move in straight lines, so you can't turn right to reach the hiding baddie".
Bobs player would most probably tell me to feck off. Now, would it be possible to use chess queen movement rules? Shure. Would it be fun? Possibly, even though I personally doubt it. Would it hinder suspension of disbelief for players used to traditional style grid based movement and lessen their fun? Absolutely, at least for the gamers at my table.
It isn't
one applying
@loverdrive's suggestion—it's the whole table agreeing to give it ago. Of course if you impose such a blatantly arbitrary restriction out of nowhere on somebody used to less-dissociated* mechanics, they're gonna be unhappy. As for "would it be fun?",
@loverdrive has already stated that it was lot of fun for their group.
A bit more generally, these were explicitly labeled as experiments, to poke at the invisible walls delimiting their gameplay and see what effect that had. Some of them turned out to be more engaging and fun; others did not. I strongly suspect that, regardless of that kind of outcome, the exercise gave
@loverdrive's group ideas for play using the baseline rules they might not have thought of otherwise.
* Remember, all game mechanics are dissociated to some degree, which is part of @loverdrive's point.
Yes, traditional grid based movement and combat is just a tiny bit more anchored in simulationist gaming than chess queen movement would be. And finicky simulationist aspirations most of us old folks got tired of in the eighties. But I don't really see what further arbitrary combat restrictions would add to a game, more that making it more gamey and removing parts of the game - such as traditional grid based movement rules - that has become transparent to many of us and thereby letting us focus on the parts of the game that we enjoy.
Making it more gamey—or at least recognizing how gamey the game is—seems to be a major part of the point. It's examining the premises of play: Making the transparent (that is, invisible) become visible for purposes of analysis and appreciation.
You could alter grid-based movement in a different way, ditching the "all directions on a square grid are measured by squares", and making a diagonal move cost 1.5x, or ⎷2 if you really wanna get precise. It's still fiddling with what are essentially arbitrary game rules imposed by using a grid in the first place. You could go full freeform and have players measure their movement with strings. And yet you still have turn-based play with each player moving and acting completely independently of the others in serial fashion. That's pretty unanchored and pretty gamey. But folks enjoy it, as you said, and most people rarely think about how artificial and nonrealistic turn-based combat is, it's so common.
A more simulationist "arbitrary" rule change might be this. Every round is split into two phases. In phase 1, all combatants move, essentially simultaneously, without knowledge of how the others are moving. (Obviously, this has practical problems in face-to-face tabletop play, which is why such a system isn't regularly used, but it could be done, and would certainly be straightforward to implement with a virtual tabletop.) In phase 2, all combatants perform whatever actions they want.
You'd have to play this out to really see what impact it has, but some are obvious. You could move to where a foe is with the intent of smashing them, but when you get there, they are gone. This is a thing that can happen in real combat, but it basically doesn't happen in full turn-based combat. You might have a situation where it looks like two combatants would cross paths, in which they should "obviously" be able to engage together in that moment, but these rules don't allow it (you could make a further change to do just that, of course, it's a nested, iterative process after all).
All this is just examining
@loverdrive's finger in fine detail when they're pointing at the moon, however. It's the intent behind the "arbitrary" mucking with rules that matters.
If you and your friends find fun in putting further arbitrary restrictions, such as chess queen movement, on your gaming - good for you! But I can't see what good or fun it would add to my table.
You could have just said this bit!
Edit: Fixed a typo.