Not to doubt youmath here, but I think it is not wothout flaws. Could you please post the assumptions and the calculations.
I am not sure what exactly you want me to clarify. The numbers come from DND Beyond in 2020, I posted them above
Key assumptions:
1. The DND Beyond numbers are representative of the gaming population
2. Class selection is independent and random.
If you play a "brign what you got" game with totally independent class choices and we assume the DNDB numbers are representative of the population of gamers it would be accurate.
The main "flawed" assumption in this is independant class choices I think. In the numbers I posted on the chances of having a Barbarian/Paladin or Fighter I was assuming a random class selection. That is probably accurate in some games but not others.
This presumes no cooperative planning (you are playing a Wizard and Mike is playing a Cleric so I will be a martial). There are certainly games where class choices are dependant based on other PC classes, but it is debatable how that would affect with a Rogue choice specifically and I have no numbers to bring that into the discussion.
Easier said: 3 out of 12 classes are at 28% of all characters. Which means that they are slightly overrepresented than anything. So an average party of four has one of those classes. Of course some have none, while some have 2.
Yes, they are over represented and I agree an average party of 4 does have one of these, but that is not inconsistent with my earlier estimate that about 50% of partys with a Rogue also have a strength-based martial with Athletics proficiency. To really prove this out you need to determine how many of those fighters are strength based and have Athletics.
In the numbers I posted earlier I used all the fighters and the 28% number and making the assumption on independant class choice you get 63% that have one of those three classes (including dex based fighters)
If you say, that a party with a rogue has less of a chance to also have a barbarian, paladin or ranger is what I am sceptical of. The question is, if those party slots are independant from each other. So do the 3 classes compete for the rogue slot at all? Or is it rather that 1 slot usually goes to the P/B/F group, one goes to monk, rogue or ranger, one goes to wizard, sorcerer, warlock and one goes to the cleric/druid/bard (notice that I did not use the arbitrary w/e/p/m groups OneD&D tried, and instead grouped them more by tank/skirmisher/offensive arcane caster/supportive caster.
I think it depends on the game, although I would generally disagree with your groupings even when not independent.
In 5E, I think some games have players that don't worry at all about what others are playing, others have PCs meticulously plan what everyone is playing to cover all the roles. Some time this is player-specific. For me I generally want to play my character and I am not going to worry too much about what you are playing, but often someone else at the table is basing their selection on mine anyway.
That said I think it is roles and not specific classes that matte. So it would not be P/B/F vs M/R/R it would be melee vs skills vs ranged vs caster ..... In that respect the Monks are almost always in with the Paladins and Barbarians as a melee-type builds. So are some of the Figthers, some of the Rangers and a not insubstantial number of the Wizards and Warlocks.
So it is like this - you are playing a Rogue so we need someone to be the tank, and I will do that. That tank could be a Paladin or Barbarian, but it could also be a Bladesinger, Monk or wildshaping Druid (and have problems filling that role at very low levels). Then someone else plays an offensive caster, and someone plays a Ranged/skirmisher ....