That isn't a good comparison, police unions have mostly different support systems and allies than other unions. Police unions are not generally helped by other unions to put it mildly.
Also, enough public support exists for both sides to try and act so which one has more "public trust" doesn't matter at all, it's about exerting power now.
Teachers' unions don't even have the power to get budgets big enough for teachers to not have to buy their own school supplies for students. Talk to me when police unions are weak enough that we can cut police department budgets back so much that police have to buy their own bullets and riot gear.
Also, enough public support exists for both sides to try and act so which one has more "public trust" doesn't matter at all, it's about exerting power now.