I operate under the assumption that the majority of cops are just doing the job for a paycheck, and not out of any genuine desire to do the job correctly arising from a professional work ethic. I get it--it's just like every other job. Only a few people actually care about what they do, and the rest know what it means to just be an employee.
Giving the footage, without the name, forces the cops to positively identify the suspect through investigation of crime scene evidence. The FaceID-generated name is just speculation from Apple.
Operating under the assumption that the cops will do the absolute minimum amount of work to clear the case, just giving the video runs the risk that they will decide that identifying the suspect is too difficult, and give up. But bear in mind that they failed to identify the real suspect anyway, and also arrested an innocent person, just because Apple said that's the person who did it.
The professional cops, who are usually not as knowledgeable about computing science as the professional software developers, may be unaware of the GIGO principle--garbage in, garbage out. They don't know how Apple assigned a name to the face, but they trust them, because Apple spends a lot of marketing dollars getting people to believe that all their stuff "just works".
The facial recognition software linked the face with a name attached to all the faces that didn't. Apple can certainly give a rate of false positives and false negatives for the face-matching, but I'm not sure they have any idea how many people are showing them fake IDs. That's something cops and bouncers would know, not ordinary retail employees.
If you were a cop, and someone dropped this case on your desk, what would you do? I, personally, would search for that name on every corner of the Internet before getting off of my ass. I'd be looking for any excuse I could find to not go out to arrest someone, because that's work, and solving property crimes, especially shoplifting cases, is not noteworthy or glamorous. But Apple is rich, so I'd put just a tiny extra effort in to find the real thief, in case I ever wanted to run for sheriff, and get them as a corporate campaign donor.
I'd probably go to one of the department's fake Facebook accounts, upload a cropped still from one of the videos, and see if their auto-tagging software produces a match. If that didn't come up with anything, I'd have to get a search warrant for the named suspect and their home address, looking for any stolen Apple stuff. Then I'd see that the named suspect does not match the video footage. I'd give the best face-shot and a short video clip to the local TV news to see if that generated any tips or leads, and then finally write a report saying that the suspect could not be identified or located from the available evidence. But I'd be polite about it, because Apple is rich, and would certainly want to read the report. And then that would be it. Because every retailer has to live with a nonzero amount of shoplifting and other forms of shrinkage, even the rich ones.