If you find a single instance of a nation where the police functions well without qualified immunity, you can at least conclude that it's not a necessary condition for police to be able to operate well.

Sure, in a different context it may be a necessary condition, if you consider all other variables immutable. But there's no need to do that. Many variables can be changed, including the legal system.

I think if you observe that your legal system is so janky that you can only have a functioning police force by making police officers selectively immune to the law, your response should be to not accept such a kludge and insist on actually reforming your legal system.