Two out of three of my DC have gone through long phases of this. As I said previously, one has been diagnosed and we've asked for a referral for the other for ASD (not because of the school drop-off issue).
Various things have helped at different times.
A TA who made it her job every day to welcome DS and have a job for him to do at the start of the day.
A transitional object for DS to get interested in when he was upset (in his case, a pair of binoculars)
Getting up earlier and giving more warning about when we were going to leave for school.
A visual schedule of the morning, from getting up to starting class.
Talking through what happens at school each morning and discovering that DS had difficulty with the first lesson. I then talked to the teacher and we arranged an accommodation for him.
Being firm and consistent that this is happening - not sticking around longer than necessary.
At one point I had to give up volunteering in an adjacent classroom because it unsettled DS.
Introducing melatonin in the evenings to support sleep has had the additional benefit of reducing anxiety and stress, making the transition easier.
As you may deduce, no single solution has been a final answer, because with ASD, transitions are always tricky, so every time there is a new source of stress, old problems re-emerge.
We've been very lucky with a brilliantly supportive school and SENCO, who have never needed a diagnosis to put in place the support our DC need, but it is very worthwhile getting that diagnosis.
Until then, trust your instincts. There's no harm in parenting them as though they do have ASD - it just means being a bit more careful to make sure their anxieties are acknowledged, which is good parenting anyway.
Feel free to PM me if you want - I come from a whole family of schoolphobes, so I'm very sympathetic to this whole side of SEN issues.