A cheater isn't evenly matched against you. No one is good enough to compete against wallhacks/aimbots, never mind that it shouldn't matter. It ruins the experience, ruins games, ruins the spirit of competition and sport.
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.
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.
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.
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.