G.,
She doesn't need 'socialization'. Check into homeschooling and the myth of needing socialization. She wants to stay home. THAT IS OKAY! Question is: Are YOU going to lack socialization if you stay home?
Planning a weekly schedule around one event (and NOT building up to it) is a much better idea - like errands! You see plenty of people running errands to fit the bill of learning how to interact peaceably and politely (by practice, not by reminder).
Check into Waldorf education - and the works/writings of Rudoph Steiner. When you see something about Rhythm, - read - .... a lot! I am trying to get rhythm in my household and it is what my boys are hungry for.
It's not about the outfit, it's not about being left behind (which really isn't a good thought (or practice) for a two year old - what do you do when they ARE going to be left behind? it's sad that she is willing to part with you going out the door already!) It is about knowing what comes next and a whole host of other things that a child values but current society does not.
She doesn't need to go to the museum. She's two and a half. Not 12! She isn't going to remember it - even if she has fun when she finally gets there - it isn't a lasting fun - each time it is hard to get her out the door, it will get harder and harder long before it ever gets easier.
My suggestion would be to cut everything down to her music class and any necessary errands. I think you'll see a lovely transition to a little girl who is happy to interact with the world as SHE is able to process - not as someone else sees her as needing to meet some invisible goal.
Less is more. and when it comes to doing stuff, more IS less. Less enjoyable. Less wanted. Less comfortable.
I have been slow to learn to pare down my visits to the outside world trapped in a car seat for them. Cutting back on gas consumption - and giving them more TIME to experience and learn and BE who they are instead of giving them discomfort of social situations that are always new and unexpected.
Visit friends. Plan a playdate one to two times a month. Otherwise, stay home and hug your child every time you think of what they are missing - because THAT is what they are missing - you - and she is trying to tell you that but really, has no words for it.
Good luck!
M.
PS: read your update. You can still create a rhythm at home - and just going to the tree on the street (if you have one), and observing everything can be an outing she will cherish and look forward to. I do agree you have to get out of the building. But you don't need a 'park' when a tree, or even a hardy little plant growing in the sidewalk will do. Or a dance in the rain in her play clothes. Or singing nursery rhymes on the front steps, or....