The NYPD states that "there were no reported collisions, summonses, injuries or arrests in regard." This doesn't mean that there weren't any, just that they weren't reported on. If a paparazzi got injury due to aggressive tactics, they would be disinclined to report as it might lead to legal liability on their part. Regardless, however, this lines up with TRH The Sussexes own statement about "near catastrophic car chase" and "near collisions."
Many may say that TRH The Sussexes are blowing this out-of-proportion, but the NYPD, who I'm pretty sure does not give every famous person in New York City personal protection (as that would be a tall order), felt that the situation called for their assistance.