I don't know if this is actually going to be helpful, but your experience really resonates with me. My DD is now almost 12. She was also a 'sensitive' and 'intense' baby/toddler/preschooler/child and every single drop off at nursery/preschool was a nightmare. Floods of tears, they had to peel her off me while she clung on for dear life. She calmed down after I had gone but she never actually found being there a positive experience. She just kind of made the best of it, having learned that once I left her I wasn't coming back for a long time.
As she got older, the tears and clinging happened a bit earlier as she didn't want to be seen crying and making a fuss by other children or by teachers at school. She'd stick on her 'game face' and go through the school gates, then basically do her best to be invisible all day.
Eventually all the years of being forced to cope in a mainstream nursery/school setting caught up with her and she crashed out of primary school mid-way through Year 6. Was never able to get back in, serious school-based anxiety and trauma.
Now having an immensely hard time trying to settle into secondary school, she's only going in for an hour and spending it with a TA instead of going into class. It is a very long and stressful battle each day just getting her into uniform and into school. She simply doesn't feel safe there, there is some deep trauma caused by many years of being expected to cope in an environment that was actually deeply unsuitable for her.
She's autistic. I knew this when she was 6 and had my suspicions earlier. She was eventually diagnosed and finally got some support at school when she was nearly 11, but it was too late.
If you know that there is something different about your child, I'd recommend a really long look at the way everything is set up for you and your family and your work. I'm on the verge of resigning from my job now, I just can't do it anymore as she needs me too much for constant support and TBH I doubt she'll ever settle in a school environment. Maybe she could have coped now if I hadn't prioritised working (mostly full-time) since she was 12 months old and kept on forcing her into childcare day after day after day. And of course school recognising her additional needs and actually trying to meet them might also have helped!
What you're going through isn't normal, and it isn't your fault, she isn't 'too attached' to you. She's communicating, in the only way she can, that she needs you and isn't coping in childcare. If your child is different in any way, if she has any additional needs, then you have subconsciously become her specialist 1:1 carer as well as her mum. You're the one who can meet her needs, and the way she feels being sent into nursery without you is the same way a deaf child would feel being sent into nursery but having their hearing aid taken away first. They've lost the one thing that enables them to function properly.
I'm so sorry, I know this isn't a helpful post. :-(