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Something like the new "select natural wonders" menu option, but for random Civs, would be perfect!
You can search this forum and find similar threads from other players reporting the same situations you are reporting: that certain civs appear in their games with greater frequency than other civs do. The interesting thing about those threads is that they aren't all finding the same civs to have increased frequency.
I suspect that the actual cause is random clustering, with some people who experience clustering perceiving it as a pattern because of the relatively small sample size of games that they have created. If all the games every Civ VI player starts were taken in aggregate there is likely no statistically significant difference in the frequency of appearance of any randomly chosen civ (assuming I am correct).
If you find that this pattern that you are observing bothers you, then one possible solution is to select the civs in your games pseudo-randomly. To do this, after you have selected your own civ, use a random number generator to select each of the other civs that will appear in your game, and then select those manually in the setup interface instead of choosing the random option. If this process selects a civ that you feel is appearing in too many of your games, simply reject it and choose another randomly until you are satisfied with the results.
In doing so you'll lose the element of surprise where you discover which civs are in your game while playing the game, but you'll also eliminate that feeling that the results are not sufficiently random.
While some of this can be chalked up to the when the other civs have been released, there appears to be a bias - some of it is due to traditional rivalries (but we all know how the Roman/Brazilian wars ended) but otherwise who knows - is it based on the inherent balance between the civ you choose and the aggressive tendencies of the AI civs? Something else?
That said, I do like your idea and I may try picking them with my eyes shut or something. The things we do to have fun playing Civ 6...
EDIT: If we dismiss this simply as the sharpshooter fallacy we also permit and accept the possibility of the same game being generated every single time. This is not likely but it is entirely possible in an unrestrained random system. That would be a very bad game and that is why there should be constrains and balances to prevent so much clustering.
It could be that the biases and weights are being amplified in some way but it is nigh on impossible to get that looked at with so much 'its just random' sloganning on the board.
Of those you have seen, I have played against both Frances perhaps once or twice each. I think I have seen Saladin once in about 20-30 games. The same with Brazil.
One persons observations.
Its just random. Play only 20 games and you might get the same leader 4- 5 times.
However, while we wait for that there are ways to take control of the situation ourselves and produce the results we want. In offering such a method as a suggestion I did not mean to suggest that the developers themselves should not take note of this situation and take action for the sake of player quality of life.
I have played many more than that and Musa has appeared in more than half.