I think that's pretty standard lease language - it doesn't mean he can enter at any time with a few police friends and search your place. There's a ton of case law about this, and there are a lot of nuances in this area of law (an entire law school course is pretty much about 4th Amendment search and seizure exclusively).
The owner of the vehicle could definitely allow police to search the vehicle. Like if JR's dad owns the car he drives, dad could consent to LE searching the vehicle. But say you were leasing a car - the dealer, or whoever actually "owns" the car could not consent to the car being searched.
Wish I could give more than 1 thanks per post I'm sure LE is pursuing all paths to get around in-cooperative POIs, and I try not to let my desperation in this case leak out to my keyboard, just really hope they are as zealous as they can be.