They still do. Although they're not the same at the barber I go to, the blade part is just a normal razor blade that fits into a holder in the arm that swings inside the handle. We had two when I was a kid that had belonged to my grandad, used to love playing with those, and the (decommissioned) WWI Webley revolver we also had from his father (i think).
This thread has just reminded me of that "Lionel Blair-cut" Harp lager advert from around 1993 which had Brian Conley in it as the hairdresser (or Barber).
Telling it almost exactly like it was so many years later - and proud of doing so!