I've been thinking about the incident at Joe's place (see OP)

There was a criminal (parolee) and he'd been beating his wife or kids. He was prob. drunk or on drugs. Somebody called for help, and the police were obliged to come and help.

The criminal jumps in a car and takes off at high speed.

What's better, for a criminal to rush through unsuspecting neighborhoods alone, or for a crim. to rush through unsus. neigh'hoods with a cop behind, flashing lights and sounding a siren?

If I'm out there, innocently driving to the grocery store, isn't it better for me to hear and see that there is danger? Because I can hear that siren from blocks away, perhaps a mile.

And if the criminal crashes, isn't it better for a cop to be right there, knowing that it happened and where it happened, and summoning help?
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And consider the alternative. If the police had the policy of "no chasing," then criminals would know that they could rob banks, shoot the convenience store clerk, or beat their families, and all they would have to do is hop in a car and speed away. Because the cops won't follow.

Beyond that, criminals don't think.