A bot can be very easily made to be effectively unstoppable in an fps, just look at hacking players.

Just as a chess bot can be made dumb as a box of rocks.

They just serve different purposes, for example in counter strike, bots are nerfed in purpose in order to discourage kicking other players to replace them with a bot that would do better.

Chess bots are also the target of computer science researchers, very few if any, serious academics are focusing their time developing a bot for a game that is either a flash in the pan as far as popularity, or actively discouraged by the arbiters of the game.