I think it’s a positive thing that your mum has been able to be honest with you, telling you the limits of what she can give (that she’s feeling like your counsellor), rather than just going along with it and feeling more and more out upon and things getting even worse.
This is going to sound really harsh, you’re entitled to feel upset and down that this isn’t how you imagined the bond between you when you became a mother yourself, but your mum doesn’t owe you any help or support. If she wants to give you that and she’s able to that’s wonderful. In my own experience only a few lucky people end up with the type of parents/in-laws who are involved with their kids, offer care and support, willing to be there for emotional support etc. not everyone is capable of that level of input. Not everyone wants to. She’s done her parenting duties in raising you and as awful as it sounds now you’re both adults the relationship has to be one that works and is willing on both sides.
You may feel you need her more than ever but I assure you, you’ll get by without her. Keep your relationship light for a while and remember you don’t know what she’s going through in her own life (and she might not even tell you) and listen to her, she’s not the person right now for you to go to overly for support or advice. Hopefully if you are in need of those things you know of other places to seek that help, whether it’s close friends, your in laws, a professional like a therapist, charities like Sure Start if they’re in your area.
Very few people end up with this idyllic scenario with loving involved grandparents at their beck and call yet it’s made out to be essential and mandatory. Families are far more complex than that. And if your mother has depression I think it’s important to recognise she’s unwell and may not even be capable of giving you the support you want right now. Obviously I don’t know why she feels like you’re asking too much of her, and she’s feeling overburdened like a counsellor, do you think you’ve been putting on her a lot recently?
I prefer to expect nothing from my parents/in laws and any help or support is seen as a lovely gift given willingly, but then I’ve always been pretty independent. They’ve done their roles in raising us and now they get to set the boundaries they work for them just like I do as an adult. Having set expectations of someone is a recipe for disaster. My own mum died a while ago sadly and I know I’ll miss her when I give birth to my first child later this year, but millions of people raise children without family support and do just fine, you’re very lucky to have supportive in laws!
You can’t help how you feel about this but you can help how you respond to her. If you’re really struggling with your feelings around this I don’t think trying to go to your mum and change how she feels is the way forward, I’d seek some therapy. If your twins are under a year old you’ll be fast tracked on the NHS IAPT service for therapy and skip the waiting list entirely.