Because you effectively end up with different coins right? Even if it doesn't break the on-chain connection with your new coins, it breaks the connection with your old coins?
Yes.
I had thought the point of a mixer was to mix your coins with other peoples, not just your own.
The point of a mixer is to obfuscate the direction of your money.
Surely this defeats the point of a mixer if you're just running CoinJoin with yourself?
It's pointless as long as third parties know that all of the CoinJoin's inputs are yours.
I get that the coins would still technically be new, but it doesn't do anything to break the connection between old and new coins if you're only mixing coins from the same origin...
The same origin needs to be proved it's the same. A bitcoin transaction can be consisted of inputs that are not owned from the same person.