“they could do it by REDUCING the police budget, and using the money for more court and jail capacity. Plus, having a policy of not bothering productive citizens who happen to kill career criminals in self defense”
More police leads to higher crime. It’s an axiom.
The reasons are myriad, but the principal reason is that criminals are not imprisoned for the duration of their sentences, and many sentences are reduced.
And this is done to pay for more police and associated technical infrastructure/equipment.
The great lie is that the Police keep people safe. The Truth is that Prisons, Harsh Sentencing and Executions keep people safe, to the extent that government can.
Police keep the ruling elite safe.
Throughout the world, the real reason for police is to put down rebellion against the ruling class. Whatever they accomplish beyond that is secondary.
Full time police are the "standing army" which the founders warned against. There were no police in the early Republic. There were sheriffs, but their power was based on their being able to call out the posse at need (and upon the willingness of the posse to respond). Criminals were captured by the armed citizenry.