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?
Correct. And correct.
I had thought the point of a mixer was to mix your coins with other peoples, not just your own. Surely this defeats the point of a mixer if you're just running CoinJoin with yourself?
It's not a coinjoin. You get different coins than the ones you started with.
it doesn't do anything to break the connection between old and new coins if you're only mixing coins from the same origin...
That's impossible on the Bitcoin blockchain. Think of it this way: Alice gets Bob's coins, without knowing who Bob is, and without knowing how Bob got his coins. Alice's old coins are now owned by the mixer, and Alice's old coins can't be linked to Alice's new coins.