Scrub the audio? I don't think he's going to go in there and literally separate and remove the song's frequency from his voice which are interweaved together. Something like that takes a tremendous amount of knowledge and time and always comes out sounding bad because any time you remove shared frequencies it'll make your voice sound very choppy and robotic (I know because I've done things like this). His job would slowly become being an audio engineer instead of a traveler.

Overlaid a royalty free musical work to replace the copyrighted one? Not possible, his voice and background audio is on the same track. To do that would essentially mean muting the entire track. This is the easiest option and often what he does. He just cuts it or mutes it, both of which makes his video content worse.

Asked the market vendor to turn the radio off? He often does this in closed stores where he plans to spend a bit of time in, but it's unrealistic to expect anyone to do this in an open outdoor market with ~200 vendors spread around everywhere for 1,000 feet in every direction. You're asking to basically shut down the whole market and control every audio source while you walk around. Not happening for a regular person.

Waited until the radio played a work that was not copyrighted? Not possible due to the above reason but this is also a little more unreasonable because there's no reasonable chance a regular person will be able to determine which songs are copyright and which are not, especially not in another country where they don't even know what song is playing. Also by the time he researches the song (assuming he somehow figured out how to even determine if it's copyright or not), chances are a different song would be playing and now he's stuck in a research loop for the rest of his life, or ends up in an another infinite loop waiting for a copyright free song.