Well no, actually.
By the time the first football winner was named - Bobby Moore in 1966 - there'd already been 4 atheltics winners, 2 swimmers and a show jumper!
Other football winners: Gazza in 1990, Michael Owen in 1998, and then finally Beckham in 2001. Two out of 36 pre-Sky, and two out of 16 post-Sky.
Cricket winners: 1956 (Jim Laker), 1975 (David Steele), 1981 (Ian Botham) and 2005 (Freddie Flintoff). Not exactly dominating the event even when the BBC had the rights.