That is what happens in Civ (civ4 at least)!
The AI bonuses are static and don't change through time, you don't get a bonus by being last or any penalty by being first. In fact, by being the leader you get more bonuses than everyone else by being able, for example, to build wonders and get religions before everyone else. I really don't see what your point is .
If anything, Civ (and most strategy games) are the absolute opposite of mario kart in that regard. If you're first you get all the better items than if you're last which results in run away growth. Personally, I hate this and find it hugely unfun. If I'm slightly ahead at the renaissance, my advantage often snowballs so that by the time I launch my spaceship the others are barely discovering oil. This ruins the late game by removing the slightest element of challenge in the game. And if I turn the difficulty up for my next game, then I'm completely unable to get the slightest advantage at any point and get massively overtaken by the AI and lose horribly far before the modern age.
Mathematically, let t be time and Fa(t) be how well civ a is doing at time t. In civ with a positive feed back system (being the tech leader makes it easier to discover new techs; having the most land makes it easier to grab even more land, ect...); Fa(t) is a roughly exponential function although there are many many factors that might make it dip or rise at any particular point, on average over say ~40 turns the growth is exponential. This means that if Fa(t) is slightly > Fb(t) (a and b being different civs) for a small t, then for T >> t, Fa(T) >> Fb(T) due to the nature of exponential growth. If you make the assumption that the game is fun when it is slightly challenging and civs , you can define the function Ga,b(t) = Fa(t) / Fb(t) which is how 'fun' the game is between a and b at time t -> 1 corresponding to the most fun, 0.0001 being a walk over for civ a and 10000 being complete doom for civ b. Thus, if Ga,b(t) = 1.1 (ie a fun game with the civs in similar power) then for T >> t, Ga,b(T) >> 1 - which isnt fun. However, if Ga,b(t) = 0.9 which is also a fun game , then Ga,b(T) << 1 which also isn't fun. This makes it very hard to have a fun late game because if anyone gets a small advantage at the start this becomes a massive overwhelming advantage at the end - which is fun for no one involved.
Mario kart had, in effect, exponential decay so that no matter what Ga,b(t) was, Ga, b(T) would be closer to 1. As you were complaining, this makes it hard to get a massive lead as it punishes you for doing well - but it does make the game more fun for a LOT of people.
Ideally, for civ Fa(t) would be a linear function, so that you are rewarded for doing well, but not so much that the game snow balls. For example, if a plays better at the start and gets a small advantage so that Ga,b(t) = 1.1 at the classical era; but then doesn't play any better than b for the rest of the game, then Ga,b(T) should still = 1.1 in the later game. This would make for a more fun civ.
You're making an assumption of what fun is for other people. Personally, I don't find it fun when a game "cheats" for the players who are behind so that they can catch up. Mario Kart got old really quick because the best player had a similar winning margin not matter how well they played or how badly other people played. It didn't take long for me and my friends to just stick exclusively to battle mode and just ignore racing mode.
Imagine if basketball had a rule that if a team is down by at least 10 points, they would score double points until the deficit goes below double digits. It would make the late game more fun in the short run, but I would think that people would sour on a rule like that pretty quickly.