The best way to stop cheaters is to program the game to be Server Authoritative with Client-side Prediction. If the server does all the math for checking damage, whether someone hit something, speed and position, etc, then the client cannot cheat those values no matter how hard they try. The server will tell everyone else the correct values and the cheater will keep getting reset to what the server says is true. The only kind of cheats you can use on a game like that would be aimbot or wallhacks. But both of those can often be detected using anti-cheat software which acts like a rootkit. So a combination is most often used.
Like I said, I don't think open source is necessary. If the server binaries are released, then people can run their own private or join someone else's by IP, just like online games used to work in the 90's. That is plenty good enough for consumers to keep using what they paid for and takes zero effort on the part of the developer. Its just the reason they don't is a combination of what I said before and the publisher wanting you to stop playing the old stuff and only buy the new stuff. Nintendo is notorious for this, and one of the reasons my strong dislike for them has been growing.