My dd was just like this. She ended year r with only crying some days but not all. Pattern continued into Yr 1 & 2 with her being fine for a few weeks then crying every drop off for a month or 2. Moving her wasn't an option as other dc was in year above & very happy.
To be honest she carried on like this on and off til about age 10 (didn't cry at drop off after year 2 though). Often cried when dropping her off for hobbies, which she wanted to do, refused to quit but didn't like being left - even if she could see me sat in the car the entire time.
She didn't like me leaving her with her dad when i was going out to meet friends for an evening etc.
No trauma in her life, I'd always worked and she'd been in childcare from 6 months old. It's just how she was - overly anxious & worried.
She refused to do sleepovers at friends houses so we'd have them at our house instead. Her hobbies involved a lot of traveling which we always had to take her as she wouldn't go with friends instead so no lift sharing.
I spent a lot of time working out strategies to help her with anxiety (printed out loads from the internet, which really helped), and it improved over time.
She's now 17 & is completely fine!! She really became more confident around year 6, would finally do sleepovers at friends' houses, stopped being anxious, and just got on with things. She became a different person overnight!
When I ask her now why she was like it she doesnt know & is really embarrassed by her behaviour. All she says is that she liked being at home. She is still a home body but has a part time job in hospitality which has increased her confidence in being in new situations with new people, still does her hobby but for a new team where she knew no-one when she started and took it all in her stride.
Looking back, there's nothing I could have done differently. She was just an extremely anxious child who didn't like a lot of things, especially change.
She's also always been extremely close to her brother who is only 1 year older who has always been the complete opposite personality wise and I think that helped her as he encouraged her to be more social & take things in her stride.
Changing schools for her wouldn't have helped, but probably would have confirmed for her it's right to not want to go as I'd pulled her out.
I was quite tough love about it all. Always reassured her I'd be back, or where I would be waiting for her, but said it once and then left. Didn't give her the opportunity to prolong any goodbyes as that only made it worse. Was very matter of fact - you have to go to school today. I will drop you off and will be waiting at the tree at 3pm to collect you etc. Have a nice day - love you, bye. Turned around & left. School were good & made sure they took her straight in with no hanging around.
The only thing I learnt about her over time was that giving her notice of things was not helpful, even though I thought it was. Instead of using the notice to get used to the idea, she would instead have longer to worry & panic over it, which made it worse.
I'm hoping it will work out for you like it did for us, eventually!! Good luck