The burglar's phone's MAC address in your router's nicely timestamped log would be evidence, I would think, albeit less iron-clad than surveillance video of him taking your stuff. It would serve to bolster the prosecution's case, should the police manage to find the perp though other means. E.g., the thief might have been found fencing your Vermeer. He might claim to have acquired it innocently from someone. In that case, the jury would find your log interesting as they weigh the evidence.

An interesting question is whether the MAC address alone could be used to trace the perp. The first 24 bits of the 48-bit MAC address identify the company that manufactured the adapter. Then the question would be, did the company that put the adapter into the phone cross-reference its MAC with the phone's serial number and the serial number with the owner.

A smart thief would turn his phone off during a job. Routers logging MAC addresses are probably a much less serious problem than cell carriers keeping logs of which phones were where when.