Mineral spirits and oils only cover the rust. What phosphoric acid does is actually chemically bind with the rust and turn it into ferric phosphate and water meaning it is actually gone. In the photo with the cup and the foam inside it, all the black stuff floating on top is that material that's fallen off the metal as it can't hold onto it That's why you use a metal brush after the soak to remove the black ferric phosphate and give you clean bare metal.
Doing it this way takes longer but removes all the rust without covering it and slowing or even stopping any future degradation.
I use rust remover on car parts and on my lawn mower I use rust converter as it turns the rust into a protective coating to prevent further rusting.