Well, you'd kinda have to if you were assigning it as homework.
We could go round in circles.

(jumping to conclusions all the time, that is) For starters, there could be a class computer, a couple of computers with it installed in the computer room, or some other arrangement.
Let me rephrase then; its perfectly possible to play Civ (particularly if you aren't very good, don't really know what you're doing, or aren't trynig to win) without doing any math.
For plenty of kids it could be mindless clicking.
True. For many kids, math exercises are mindless manipulations as well. Surely there's always a risk some kids won't learn as much as others do. A task involving civ certainly doesn't need to be assigned to someone who has zero interest in the game or who clearly could not get any benefit from it based on their current level of development.
<I'm a bit tired now, so this point is probably not very clear and I'm leaving myself wide open to counter argument, but such is life.>
Sure.
But when I was in high school, I think teachers generally had kids play "learning" games like Civ or SimCity or Diplomacy (boardgame) or watch a video as an excuse for having a real lesson plan. It was a way for teachers to have a low-stress class where they could sit around and catch up on grading.
I (and most others) learned nothing from these.
Ah interesting. By high school, I think the best we ever got to do in spare class time was minesweeper or solitaire.

I am honestly surprised you flatly say you learned nothing from these. Particularly with the board game, and assuming you didn't all sit in silence playing civ or simcity individually, I bet you were all talking with each other. I'm sure you know this (don't want to sound too preachy) but students learn arguably more effectively from each other - peers - than they do from teachers. Whether it's a cultural, generational or language gap - whatever you want to call it - it is real IMO. No matter how "with it" I think I am, I simply am not.

Anyway, those easy lessons may have had little obvious educational value in terms of the outcomes for the subject and so on, but can you deny they had
any value? Even schoolchildren deserve occasional breaks. It shouldn't always be viewed as a waste of time. Are they not building relationships etc. etc.
I find it hard to see how you could incorporate it into just a single lesson or two. For kids in particular who don't have prior experience, its going to take quite a while for them to even figure out what is going on, more than just a part of a single lesson. Let alone play much of a game.
It's true. Probably would be difficult to put in a single lesson. I'm sure there would be ways to work it into a fun assignment (for maybe 12-14 year olds), where there are other alternatives that are similarly fun. The project might require playing through at least one game and giving a report on how a particular math topic is related. It's a fact that these days there are a lot of "geeky" and very computer literate children who are pretty young. I bet many would jump at the option of getting to play a game as part of a "fun" assignment.
As for a single lesson, maybe you could build a mod that is a heavily restricted scenario such that it is not as epic as the normal game. Remember civ4 is incredibly moddable. However now we're talking about a teacher spending a lot of time building a mod.
I am guessing that you had more of a background with computers than the median/lower quartile student, and were of generally higher ability than the median/lower quartile student.
Probably but not by much. I didn't know how to do much with computers, but the family was lucky to have more than one computer, so there was always a computer one step down from the business computer that was available for us to play games. I think civ1 was loaded up through DOS, and I still have memories of how confusing the DOS console commands were, which my older brother seemed to know how to do.
I was probably most different in that I was a bit of a numbers kid at an early age, as was my older brother. I remember he had worked out with pen and paper strategies for civ1 that were shown to me as if they were gospel. Things like putting a phalanx in every new city. What city growth thresholds there were based on which buildings the city had. etc.
My ability to grapple civ1 way back then would certainly not be exceptional by today's standards. Heck we didn't even have the internet back then
I played Civ1 when it came out, but I'm old.
So you have wisdom to share. By the way, I have such fond memories from around the time I played civ1, it's part of the reason an iconic part of it is in my avatar.

I remember specifically getting to go home 5 minutes early, along with a few other lucky kids, from a class in grade 2 (about age 8 I guess) because I was able to think of a word starting with 'ph'. Can you guess what it was?

It's funny actually, the teacher didn't know what it meant, and neither did I really, but she seemed to believe my explanation that it was an ancient military unit (I didn't even know it carried a spear). I probably freaked her out that day.