I held back both my boys - summer b-days in late May and July. My oldest went to kindergarten at a public school at age 6 and it was the best thing. He was not ready emotionally or socially and academically he needed another year as well. No regrets.
My youngest will start kindergarten this fall and the same public school and he will also be 6. I just think it gives them that extra time and you can't ever get that time back. You can always do the private kindergarten at your preschool and depending how he does decide whether entering kindergarten or 1st grade at the public school is right for him. Kindergarten is completely different than when we were kids and the expectations are increased. What I did in 1st grade is what they are doing in kindergarten now, so keep that in perspective.
Best of luck and go with what your gut tells you.