Although the best option would either be your own cold storage or a Passport hardware wallet, I would definitely still keep your coins on your Ledger over a centralized exchange.

I think the best way to approach this would be to initialize your Ledger, update it, and install the bitcoin app, all via Ledger Live as normal. Make sure you don't go anywhere near enabling Ledger Recover. Once you've done that, uninstall Ledger Live and use your Ledger via some open source third party software such as Electrum or Sparrow. Although not ideal, doing this reduces the risk from Ledger Recover to a minimum, and it is still far safer than using a centralized exchange.