OP, when your dd has her out of school social life, presumably you do things to help it - take her places, or have people over, phone people and so on. You have to smile and them and talk to them and be polite to them and presumably you don't look aghast if someone you already know from your street or existing friendship group or family say to you something like "what about Wednesday?" or otherwise suggests some way of getting together? Presumably unless all your socialising is with family there must have been a time when these people were new to you too?
I just can't quite work out why what you must do elsewhere to grease social wheels becomes 'bullshit' when done at school, unless you've got a really low opinion of all the other parents at your dd's school.
I do agree with you about your dd being young to go with a parent you don't know and I would probably say no, but I don't think the other parent asking is wrong or that it's something you can just ignore and not take part in at all without it having a negative effect on your dd, really.
She's right at the beginning of her years at school and I think you might be surprised at how things change over the next few years. However great her out of school social life is, having even one or two school friends that she sees after school will probably be a good thing for her and make her feel happier and more confident at school.
Encouraging any blossoming friendships with after school playdates and so on will take you out of your comfort zone a bit but it could really be worthwhile for her, even if it doesn't seem like something she's missing out on much right now.
You can always tie her shyness to her age so it comes across less of a permanent character trait as to do with her being four.
I'd apologise for being caught on the hop, say that at four you feel she's still a bit young to go by herself, but (IF your dd has said she's keen) how would your dd's friend like to come to your house for tea, with her mother. I know, I know, it sounds like hell . But given you can't invite yourself to their house without coming across as a bit pushy, it's probably the only way to nurture rather than reject the friendship overture, without having to send your dd off alone.
If they go for it, there you are, investing a little in a potential friendship for your dd, and it gives you a chance to check out the mum so that's one parent ticked off your list for next year or the year after when the issue of going alone will come up again and it won't be so easy to use age as a reason not to let her go.