>made a lawful attempt

Basically a meaningless distinction here - almost anything cops do is considered "a lawful attempt" simply because they're cops.

That doesn't mean it was the right thing to do, or even that it would still be a legal thing to do if we had a justice system that held police to the same standard as everybody else.

>How would you have apprehended him?

Slip an air tag in the car or hit him with one of these, then track him down later. The car thief didn't pose an immediate danger to anyone until the cops got involved, so there was no reason for them to create the situation which resulted in his death.