I have mixed feelings on this, if I'm honest.
As I have no idea whom the player concerned is, I'd say that unless his 'family man' image is something he has chosen to tout about to increase his standing/profile, the only people who should be be concerned about his antics are him and his wife/family.
If his 'image' is what matters, especially to his sponsors, and the public who have chosen to endorse him because of his public family-man role-model image, then perhaps he should be named.
Any of these 'aspiring starlets' ought to have more self-respect and dignity than to flog their tawdry antics to Fleet street ........and the great British public ought to have more class than to want to read them.........however, to use the law as a blunt instrument to gag these 'starlets' using a wealth that we can only dream of sets up a dangerous precedent.