Kids do pick up on parents' anxiety, for sure. It's tough to know whether his separation issues are fueling your anxiety, or whether your anxiety is fueling his separation issues. Or both.
It sounds like he did well separating after a while - a few tears and then he was fine. That's a normal progression.
He may also not be ready. My son was very social and did well in groups, but I did not have him start kindergarten at 5 because he had napped well past the age of 4 and I didn't think he could handle the afternoon program. He went full tilt with a lot of energy and then napped from 1-4 PM, so there was no way afternoon kindergarten was going to work. I gave him another year of preschool (sort of a pre-K) with a lunch option 3 days a week. He learned to transition from one program to the other (change of teacher, change of some kids, change of room). Then he started kindergarten at age 6.
It had NOTHING to do with intellectual development or academic skills. It rarely does. This was the best decision we ever made.
Since then I have taught in several schools and I can't tell you how many kids really need to repeat kindergarten because they just aren't ready for first grade. It's a maturity issue. If you say you are "freaking out" then I think you need to look at your own nervousness as well as the fact that maybe you just know in your gut that your kid is not ready for kindergarten! Do not think for one second that you're not letting on that you are nervous! He is picking up on it big time. There's no benefit in rushing a kid into kindergarten. If you get any pressure from friends/family, IGNORE THEM!!
Is there a a reason you are sending him to private kindergarten instead of public school? Those reasons need to be factored in too. If you don't mind paying for private education, then figure out if he should go to this school for one year and possibly transfer to the public schools, repeating kindergarten, or if you plan to keep him in private ed on an ongoing basis. If you are willing to pay, then maybe he needs a year of Pre-K instead of just summer camp for a few weeks.
In public school, you will not be going into the classroom to drop off though - he will either take the bus or he will be dropped off by you in a drop off area and he will go into the classroom.
No matter what, he should be going into an orientation program (public or private) where he meets some kids beforehand, and he should not be going into a room with completely unfamiliar faces. In public school, it will be kids he knows from the neighborhood, plus those he meets at "get acquainted" day.
The rest of this are your issues which you need to work on - figure out what frightens you, why you think he isn't able to make friends, and how long you think you need to accompany him all the time.