You make a few wrong assumptions about how bureaucrats are appointed in Canada. I’ll speak about Ontario municipalities, because that’s what I’m most familiar with, but I believe the process in most other provinces is very similar.
For municipalities with their own police, the chief is not appointed by the mayor, they are recommended by the police board (made up of council members (often including the mayor) and civilians) and are approved by the whole council. Similarly, if they are to be removed, the mayor would not be able to do that on their own, they would need the support of council.
Also, it’s not just the council who has oversight. Municipalities are a creature of the provincial government, so they can overrule a municipality. This process may seem similar to what can happen in US states, but the difference is that it’s overriding the decision of municipal politicians (to not remove a police chief), rather than removing a duly elected politician (impeachment).
I know it’s hard to see things a different way than the how it’s “normal” where you are (I have a few personal experiences doing this), so I can see how it might seem that politicians could corrupt these bureaucrats under them. However, as a Canadian, I can say it’s pretty great having an independent judiciary, elections administrators, and even police (to the extent that police can be great). There have definitely been issues with politicians trying to force through friendly appointments (we had a controversy just over 5 years ago with a nomination to lead the provincial police), but it’s generally worked out fairly well. Since appointments are within the system of governance, there are some “checks and balances” which I know Americans love to tout.