This argument is that the match-making software is incompetent. If what you say is true, and the match-maker could determine skill with any ability, then the cheaters would quickly find that the only people they match-make with is other cheaters. The non-cheaters don't care that the cheaters exist if they never play against them.
The problem here appears to be the banning. If the cheaters are never banned then they will continue to only play with other cheaters, and everyone is happy. And in fact, to a normal player I doubt they care very much if the player is legit and smurfing or if they are not legit and cheating. That player ruins the game they are in.
The ranking system needs to be a better determinate of skill, especially early in a new accounts life, so that they can stop harming normal players games. This might mean changes to the rules of a game to allow this to be done better. The match-maker should take this into account, so that if a player does go up against a player that was far from the skill level that they end up at, it should protect that account from being placed with new players for a time so that they can forget about it.
For the example you choose for Chess, you might force players to do Chess Puzzles before they can queue for their first match. A normal player would then never see any cheaters.
Cheaters don't want to play against other cheaters. If they end up against only cheaters that's a kind of soft-ban or shadow-ban and once they figure out that's the case they'll do the same steps as if you had actually banned them. It also angers legitimate players to know that the top ladder tier is for cheaters only. If you're 200th in the world and legitimate, other players will say you only got that rank through cheating.
And the very best cheaters are still good at the games they cheat in, they just want to use cheats to be even better. One famous example in a game I play is Riolu in Trackmania. He was probably one of the top 10 players in the world. But he wanted to be #1. When he was accused of cheating it took a mountain of evidence for anyone to believe the accusations because he could set a world record live in-person. He just used cheats to be able to do it with fewer attempts.
Riolu is a uniquely terrible example. While he used Cheat Engine to slow down gameplay, he could have just as easily used TAS to record and replay his inputs since TrackMania is deterministic. This is still possible today. This will always be possible even with Kernel level anti-cheats.
I'll note here that the work that Nadeo has done on the matchmaking aspect is in line with what I'm thinking and should be expanded throughout the online gaming space. A division 10 COTD player will never see a cheater. If cheaters do show up, as they commonly do in Weekly Shorts leaderboards, the community ignores it. Their region leaderboards do a much better job than typical games of bringing the community together and they promote continuity. When top players smurf COTD on a new name, the community sniffs it out within the hour. TM doesn't need anti-cheat.
Two reasons this doesn't work
Most competitive games these days are free to play. A cheater gets banned, makes a new account, and gets placed on the competitive ladder level of a new player, and stomps their way up the ladder against unskilled players until they get banned, and repeat.
For players that cheat less egregiously and don't get banned, it's still obvious many times when someone has no skill but is using cheating as some form of assistance. It's not fun to play against a player who has a similar K/D ratio as you because they suck at aiming but can see through walls, or because they can instantly headshot people but have bad positional awareness or understanding of other game objectives like capture points etc.
It's like telling a high level chess player that playing against a child with poor chess knowledge but they're allowed to just ignore checks and flick your pieces off the board is similar to playing against an equally skilled non-cheating player just because they're capable of beating you only 50% of the time. A victory doesn't feel earned, a loss doesn't feel like an actionable learning experience.