I still believe in the ideal of a government of, by, and for the people. As time goes by, we get further and further away from that ideal. That doesn't have anything to do with the merits of the ideal.They don't serve anyone, they work for the government. The people who hire them, supervise them, make the laws they enforce etc... They do not work directly for the public and absolutely aren't in a servant position to the public. The public doesn't pay their salaries. The public pays taxes. The government then determines what happens to those funds.
The whole public servant thing is as silly as thinking we live in a true democracy. Nope, we elect people and then expect them to do the job they're elected to do, we don't get a say in every decision.
You are right, those services answer to the bureaucracy.
I'm not going to say "I pay your salary ' when I get pulled over for speeding, but at the same time, I believe in the ideal that they are tax payer funded servants of the public good.