Disconnected from the other school mums

I would love some help with this:

I’m feeling disconnected from the other mums at my daughter’s school. It’s her first year of school and I work full-time. We have a nanny who does some of the school pickups and my husband does some and I do very few.

I dropped my daughter off this morning and stopped to chat with them before realizing they were all going out for a coffee. It seems like a regular thing but they didn’t invite me.

I make the choice to have a job but I do feel resentful sometimes that my job has prevented me from being present with my daughters first year of school and also from creating connections with the other parents.

C: Other parents walked off in a group towards the coffee shop
T: I’m disconnected from them
F: Heartbroken
A: Resent my job, try to think of ways I could connect without coming across pushy or awkward. Remember all the difficult times I had when I was at school when I felt left out and disconnected. Worry that my kids will be left out of play dates because I haven’t put in the effort to connect with other parents. Worry that it’s too late and that I’m already out of the loop. Feel annoyed because I don’t have time or much desire to spend my time socializing at this point of my career. I’m afraid because I have many years of school ahead of me and I might feel this way for a long time.
R: I have an unhappy day

I’m trying to just sit with ‘heartbroken.’ I’m trying to create connections in my mind with my thoughts, to feel connected anyway but with little success. I also ignore the situation by not going to school drop-offs and pick-ups and avoiding asking the other mums for playdates because I’m afraid of rejection.

Any thoughts or advice on this would be appreciated.