Have you read Friendaholic by Elizabeth Day? I'd recommend it; a deep dive into different friendships over time, with different perspectives.
It's interesting you say in your op that you feel left out as a FT working mum. I often felt the same as a SAHP.
I guess the main thing to try and do is to take off those labels and focus on the individual. Do you have much opportunity to meet new people at the moment if your life is so hectic?
I made a new close friend last year and we just clicked and have real chemistry. But you know what? That is SO rare imo. I'm a sociable extrovert who networks and meets new people, acquaintances, colleagues etc all the time (or did when I was working, up until spring last year). I go to exercise classes, book club, volunteer at a garden and in a school, take my kids to sports training sessions and matches, meet up with local friends for walks and drinks in the pub etc etc But finding someone to have a deep connection takes chemistry and time imo.
We probably felt more deeply connected to our friends in our 20s and possibly 30s because we had a lot more free time to devote to those relationships. And if we had kids in our 30s, well, then meeting other new mums is incredibly bonding. Being vulnerable together creates that. Perhaps a new way of looking at it is to stop looking, a bit like when you stop looking for love it happens more naturally. Just be you, keep socialising and keep being vulnerable...