Fuck your car, the cops aren’t gonna find it. They’re there to write a police report so you can get an insurance claim.
The only way they recover your stolen car is if the person who stole it joyrides it and abandons it. Your neighbors testimony isn’t helpful in this scenario
If your car was desirable and actually stolen for parts or to be resold it’s long gone and the cops will do absolutely nothing
Similarly if your home was burglarized. Look at the statistics of solve rates for these crimes in your city. If they’re anything like mine they are shockingly low and the only reason they aren’t 0 is because of literal happenstance like the above scenario and not actual “detective work”
However, your neighbor opening their door for the cops does create a host of potential issues for your neighbor. Perhaps they have drug paraphernalia in plain sight in their living room? Now the cops have probable cause to search their home without a warrant because you think those noble officers are going to search high and low for your shitty car (they aren’t). Or they’re an illegal immigrant, the cops are racist, and see an in to send ins later that day. Or your neighbor is rude and standoffish, escalates things with the cop (who are known for their expert de escalation skills) and then ends up getting arrested for “obstruction”
And means you no longer get support