Yes the Ledger Nano S works correctly with Ledger Live
Also, I don't have Ledger Live open when running Electrum
...
I tried opening just the Ledger wallet, and it still couldn't find the device
If the Nano S is working fine with Ledger Live, then it would appear that the issue is specifically with Electrum and it's Ledger plugin. You might want to raise an issue on the Electrum Github Issue tracker:
https://github.com/spesmilo/electrum/issuesNote that the Devs will require that you're trying this on the latest version of Electrum (currently 4.1.5) before they offer any support.