Here's my take on it. Anxiety is tricky. You have to be able to read your child objectively—hard for a parent to do, but you know your particular child better than the teachers do even if they've had years of experience. With anxiety, you have to nudge the child just outside of his comfort zone, but not so far that he will be unsuccessful and freak-out/melt down or at least not unrecoverably freak out. However allowing him to completely avoid the anxiety-provoking situation is just as bad. They both will exacerbate the anxiety and make it build and be much worse. Avoidance sends the child the message, "Whew, good thing you avoided that scary situation. That was a close call. You should never do that again." He can begin to build up a fear of the situation and come to completely dread it. Dd1 used to do this with brushing her teeth as a preschooler/toddler among many, many other things.
As a parent you need to be able to read your child and know or make an educated guess about when to give him that extra nudge and how hard to nudge. Often you need to find a middle ground. For example, your child really wants to be able to swim, but is deathly afraid of getting in the pool. You do it in baby steps. First you have him sit on the edge and just dangle his toes in the water and watch. Then when he's comfortable with that you gently but firmly nudge him along. Maybe he sits on the first step of the pool, then the next, and the next until he's all the way in. Then you have to work up to putting his head under and learning strokes to swim. You can't just throw him in the deep end and expect him to deal and you can't let him hang out way over by the ping pong table and never even put his toes in. The hard thing about school sep anxiety is often the school is not super flexible and it's harder to find the middle ground.
Many kids do exactly as Peony and Alenushka describe and cry for 5 minutes or less and then are over it. However, some are like my dd1 and cry for hours if you do a drop and run and are still snuffling when you come to pick them up. My dd2 had some sep anxiety. It was more than most other kids in her pre-K and K classes, but did not begin to touch the degree of sep anxiety that dd1 had. I don't think any of dd1's teachers (some of whom had taught for 20+ years) had ever seen sep anxiety like she had.
For dd1 what I did was set up some playdates for sure. She loved going to other kids' houses. And then for the school routine we tried a number of things. Sometimes Daddy would drop her and sometimes Dd2 and I would. When Daddy did he hung out for a pretty decent amount of time (maybe 20 minutes) and then dd1 drew him a picture at the end. He said she was so sad and teary (though not out and out bawling) when he left it just broke his heart. We had already tried the drop and run after we got the advice that she'd be over it in 5 minutes if we'd only do that. She was not that kid. (Still isn't. I mean, she certainly doesn't need me to walk her into class any more (she's in 6th grade), but if she's really upset she just can't be done and over it in 5 minutes.) What ended up working for me and dd1 was for me to stay around through the free period in the morning, fading into the background and letting dd1 integrate herself into the activities, until they got ready for circle time and I would stay through circle time and after that I would leave. This ended up being nearly an hour or so. As she became comfortable with that I began to excuse myself during the free period and say I was going to take dd2 (she was 2.5 at the time) outside, but I wouldn't leave yet and I would come back in before circle. (Super important not to sneak away, again makes the anxiety worse.) Sometimes I would tell her I needed to talk to another teacher or administrator. Basically I was stretching out the time that she was on her own in the morning, but leaving her a security blanket with the knowledge that I hadn't really gone yet. As she began to make friends and enjoy the activities it became less and less necessary for me to be in the room. I would often be outside or elsewhere. I also began to leave at the beginning of circle time rather than the end once she became comfortable with the routine.
Her separation anxiety continued for several years. Kindy and preschool were definitely the worst, but she liked the long goodbye in one form or another for a long, long time. Later on when dd2 was big enough to attend the same school I would drop dd1 off and then take dd2 into her class and settle her in and then come back a second time to say a real goodbye to dd1 and then say goodbye to dd2 and go. I was advised by several people to just say goodbye to dd1 and not come back a second time, but they didn't know her. They thought it would make it worse, but she was happy as a clam when she knew I was coming back for just a real quick hug and kiss. It was what worked for her.
Dd2 was different. She definitely did have sep anxiety, but not as bad as dd1. What worked for her was the whole getting settled in and then I went to say my goodbye to dd1 and then came back to dd2 to say a second time to say goodbye to her at the "goodbye window" the school had for K. You could open the window and get a hug, but didn't go back in the classroom. That window made dd2 sad for quite awhile until we realized that she really liked it when she went right into her 1 minute "reading meeting" and 1 on 1 attention from her teacher immediately after we said goodbye at the goodbye window. She loved just that smidgeon of structure and then she was really okay with the rest of the day. No more big wails and sobs. So for her the middle ground was nowhere near as elaborate with as many baby steps as we had to make for dd1.
You know your child best. You've got to nudge him a little and find that middle ground. Getting to know the other kids through playdates is super helpful and is probably the easiest thing to control that you can do to help him adjust. See how open the school and teacher are to you hanging around a little. Maybe you could volunteer. I think it would be best if you didn't engage with him much, but were more the security blanket/nightlight in the background. Maybe you could cut out materials for class or something like that. Or maybe he's one of the kids that really does do better with the "yank the bandaid" method and will only cry for a few minutes after you leave.
Best of luck!