What happens if I delete the padlock from the list's name on my device?

Nothing changes — the list stays locked. The padlock in the name is only a label; the real lock lives in Relay. Editing the name on your device can't switch it off, and on the next sync Relay notices the padlock is missing and quietly puts it back. If you actually want the list to start pushing to your task apps again, unlock it in Relay: click the padlock next to the list there. That's the only switch.