Sostratus
Deity
I've played over 3000 hours of 6 now. That's a pretty good sample size to draw on. Or would be if I wasn't playing Marathon 99% of the time
So there realistically is only 50-60 odd games and in that I still have absolutely had more of some Civs than others. I would expect to see that gradually flatten out at two to three times the number; though the different length of time leaders have been in the game likely messes with my perception somewhat too.
Just to demonstrate this effect: a quick graph of one particular roll of the random permutation generator in matlab. (I used the randperm(n,k) function.)
Suppose you played 42 games, one for each civ. You only saw one civ twice in those 42 games, and some civs you saw 10-11 times! Maybe you play until a round 100 games. Things are a little better but some civs are in 20 games while that sad civ is at 6. That's over a 3x difference in appearances.
If you bump up to a standard size map, and play 100 games (guys, seriously, this is so many hundreds of hours...) we see that the last civ on the list was rolled 33 times- about twice the average. This guy is @labellavienna 's Shaka. If you played yet another 100 games so you had 200 total standard maps, we can see that things start to even out a little towards a uniform trend. But even here, "Shaka" has managed to poke his nose into about 51 games where a couple civs are around ~23-25.
The overwhelming majority of civ players will not play 200 standard size maps to the point of meeting everyone in the world. So even in the entire life of the game its quite unlikely that you wouldn't notice some popular civs.
On top of that, to really trigger your cognitive biases that all humans have, all that needs to occur is you see someone in a couple sequential games. And that is basically guaranteed to happen if you play enough.