We are speaking of public places. Obtaining a permit for commercial photography does not changes the statute of the place or make it private. Everybody including "residential associations or busybodies" may still walk along. And that is fine.
Though photographers may be tempted to say "I paid my fee, so get out of my picture."
Absolutely they may walk along, what I was pointing to was more of the 'you cant stand here' mentality that you referred earlier. Busy bodies might determine that 'you cant photograph here' and unnecessarily bother you.
It's not to make the public park a private space as much as to keep it 'public'.