No brake pads to replace once a year or so, no rim replacements every few years -- but their rear tire needs to be replaced (or at least rotated to even out the skid pads) every month or so.

The "less maintenance" bit certainly does ring true for ditching gears and going to a single speed or a fixie. But as for ditching the brake -- no, that argument does not work anymore, because the maintenance you save on your brake is greatly exceeded by the additional maintenance required on your tires.

Ditching the brake on a fixie is a fashion decision, not a "less maintenance" decision. The only thing the brake will hurt is if you do bar spins during tricks -- but even then you could just get a detangler.