No -- the adaptor's MAC is an essential part of the transaction, while cycling MACs would be a dead giveaway and would increase attention paid to that system and its travels.
Turning off the adaptor is the only meaningful way to avoid tracking.
> Say a burglar breaks into my house with his iPhone in his pocket. Could I later prove it was him by pulling up some log on my router that was picking up MAC addresses going by?
Yes, but only in a society that would allow this kind of tracking of people, each of whom is presumed to be innocent. Usually a person is first identified as a suspect, after which a technical track can be made. But a person who is not already regarded as a suspect can't be (legally) subjected to this kind of surveillance.
> And why isn't there some software (to my knowledge) that does the same thing for surveillance - logging all the MAC addresses and creating alerts if a new one comes into the area?
Because this is privileged information having to do with privacy, and violating it would confront certain well-established civil rights that vary from country to country.