Is it actually possible to connect your own node to a public electrum server?
Or does that not work because the Electrum server itself is a node?
You cannot. Electrum servers are not the same as Bitcoin nodes, they have different protocols and are not compatible with each other. Electrum wallets can connect to Electrum servers, and Electrum server uses their own instance of their Bitcoin full node to be able to retrieve information.
If you value privacy, you should run a full node yourself. That, or don't use Electrum specifically.