Even more ideally, the display would also have a built-in Bluetooth controller that stays active regardless of the USB controller's attach state, such that Bluetooth peripherals could be paired to the display itself rather than to the OS (i.e. you'd manage the pairings through the OSD of the display); and then these devices would be presented through the display's USB controller as always-on direct-attached USB devices — much like VM hypervisors present host-attached Bluetooth HID devices to their VM guests. (As a side-benefit of that, as long as the computer's BIOS understood Thunderbolt well-enough to display anything during boot, then even Bluetooth peripherals would also work during boot.)
I have AirPods but will only pair them to my phone.
(I'd guess their newer models work similarly but I haven't confirmed.)