Like all things in life, nothing is cut and dry. It's not wrong to both support unions, and call for the reduction in power of police unions.
Public sector unions "bosses" are the people and politicians, so even though these are "private" unions, their alignment is to protect the political class first, cops (members) second, people third. By focusing on politic over people (bad looking cops = lost mayor, DA political race) their primary interested is not public transparency (where the money comes from, leading to lack of consumer alignment on service rendered b/c cops are serving political interests more than they should).
In a private union, the first people they are protecting are their membership (lot could be said here about how that works in practice), but their transparency problem isn't regarding factory deaths or issues at the work place because that is precisely what they are created to bring awareness to. If Ford's cars are killing people, the union actually wants to bring that up since it can hurt employment opportunities for future members.
> Collective bargaining evens out the power imbalances between workers and their employers. It doesn't really matter if the employer is the government or a private company. I'm not sure what point you're trying to make about collective bargaining and democracy since the two are unrelated. There are other ways to handle public sector unions. For example, in Canada the military is not part of a union but their pay is tied to other public sector employees who are unionized. This makes sense since you don't want the military going on strike. When the union employees go on strike to get better pay and benefits the military also gets the same increases. You could do something similar in the US by tying police salaries and benefits to another public sector group like teachers or nurses. I have the feeling that this would be extremely unpopular with the police