Law 8, part 1, Kick-off states that the distance is required at all kick-offs, including those taken after a goal is scored.
Other than for kick-offs, Law 10, part 3, Kicks from the Penalty Mark also requires that
During kicks from the penalty mark
Only eligible players and match officials are permitted to remain on the field of play
All eligible players, except the player taking the kick and the two goalkeepers, must remain within the centre circle
In practical terms rather than in the letter of the law, the 9.15 metre radius of the centre circle is the same distance as required by opponents from the ball at a free kick (and of all players except the kicker at the penalty kick).
The centre circle provides a useful means of recalibrating the referee's mental ruler used to judge this distance, given the variation in shading, colouring and growth of turf that can make this judgement less accurate.