Some of them have phase-locked loops to sync the output frequency with the mains, but probably won't stray as far from 50Hz (or 60Hz) as the mains themselves seem to do. If they do, unplugging the mains and using the battery is probably enough to get pure 50Hz back!
Only if you don’t have mains power anywhere near the recording device. You don’t need to be plugged into mains power to have it be present in a recording.
I remember having this issue when working in a recording studio during college, even when we switched to battery and turn the mains off, we'd still get hum in some guitars.
Funny enough I was recently reading an interview with producer Michael Beinhorn and he mentioned having some "mystery EMF/RFI event" happening in New York around 1997, coming from a specific block, and he had to relocate a recording session to Los Angeles because of how strong it was interfering with the guitar amps and other equipment [1].
[1] https://gearspace.com/board/interviews/1385579-interview-mic...
FWIW, I've been an ACLU member for quite some time.
FUD is only a bad thing where there's no motive, precedent, and reason for those three - which is hardly the case here.
Like, "This new roommate of yours might be a killer. Sure looks like one. And who can trust a man that doesn't have a Facebook page?", is bad FUD.
But "Two ex-girfriends of this new roommate of yours have been found murdered in the woods, each a few years after they split. And I saw rope, duct tape, a set of hunting knives, plastic gloves, and a bottle of chloroform in the trunk of his car. Plus, when he got drunk he went on and on about how he likes this Dahmer tv-series. I think he might be a serial killer and you should be careful around him", is very reasonable FUD.