Considering that, at least in chicago, the police requested federal assistance, it seems like the pushback is more of a political game played by politicians than the reality on the ground.
The police union did, but in any case, for very good reasons of federalism, the normal and proper route for such a request for federal intervention in local civil order other than as a result of open rebellion by the State government against the Union is from the Governor of the State (who will consider the need for such a request in light of and assure coordination with any state response available and in use, such as state-level law enforcement, National Guard, etc.), not from the local police command.
(I don't know, I'm genuinely asking.)
The mayor is well aware of the practice being employed, and this is a way of saying it is distinctly unwelcome without minimizing the issues being used as a pretext for it.