Yup, agree with Mrs DV I'm afraid. So much so that I'm not at all surprised to see how your responses have evolved OP - from lovely calm measured 'I am a reasonable person' OP to snipy follow-ups when some posters get under your skin a bit.
Nobody thinks you want to get back with your ex. It's not that - and that's not what people mean by 'playing happy families'.
When you have children and split up, the biggest killer is that you can never 'have it all' again. Or, at least, the only way you kind of can is if you manage a situation like this - but, it's not real and it doesn't ever work for long. You can have a new, wonderful partner who loves your children and you can build a new family unit with someone you actually want to be with - but the flip side of that is that that's exactly what your ex should be doing too, and usually does.
So you move on and your own life is good, but at the same time you have to deal with your child being part of a new family unit which doesn't include you. Meeting a new woman who is the 'mother figure' in that family, just as your new partner is in the 'father' position. Eventually, new half-siblings you don't even know, new extended family and your child possibly calling people you've never met Granny and Grandad. Etc.
It's the way it has to be if everyone is to be happy, but it's fucking hard, and it's natural at this stage to be relieved if you see your ex not making those steps, and choosing to stay 'within' the old family group. Whether this particular occasion -lunch on Mother's Day - was unreasonable probably depends on how you view that occasion. The gf wasn't being unreasonable, and she may have come to her final conclusion based on a lot of other things about their relationship, so whatever - she's chosen to move on. But it's fair to say that she isn't wrong when she's looked at this and thought - but, this isn't about his child, it's about maintaining the idea of an intact family that he should be moving on from, and can't yet. He's not ready.
I can hear you jumping in to explain that no, it's good for your child to see you together blah blah and yes I agree with that. But I would say - as others have - that that should really be focused on HIM - times at soft play together, celebrating HIS birthday together, sharing enough everyday stuff at pickup/dropoff to make your child feel secure and that his parents like and respect one another. A grown-up meal together, for YOU, as it's Mother's Day, where your child is there but the focus is you? Slightly different. The GF has decided that's where her red line is - that's fine. She might be a totally unreasonable and demanding partner that he's well shot of. Or she just could be pretty perceptive and has made the decision that your ex is not ready to separate himself from his first 'family' yet.
People on here are perceptive too. Many of them have been through what you are going through. It is hard. But -
I hope that the circumstances of the end of this last relationship, and the decision my ex made bodes well for him choosing a secure woman for his next girlfriend
Nope, you're doing yourself no favours there. One thing you are not is secure over this. You're delighted and relieved that he chose 'you' and some of the more painful aspects of moving on are not yet happening. They will, though, so get ready.