He's wrong.
OP EXH - I get that you are happy with your new relationship, and that is great news. For YOU.
For your DD it's a matter of stress, worry and concern. She's nervous and probably feeling guilty, confused and scared about meeting someone in your life who isn't Mum.
We don't know if this relationship is going to last, now that you are living together, things have changed for you both, and recently too. Irrespective of how long you have been involved.
please consider this from at these perspectives:
If there IS a long term future in this relationship - where is the fire? is this just to make YOUR life easier? Be honest.
Is this you trying to cement a fledgeling relationship, (or one that has been long standing but is only now 'official') involving a 9 yo when she is not sure about any of it WON'T strengthen your relationship.
Are you being pressurised to insert this woman into your DC lives? That is a red flag. If this is her curiosity or need to be important, I'd consider binning her on this point alone.
A new relationship needs to be nurtured. childcare (and I speak as a devoted mum) is drudgery. Being an Adult is WAY more fun than being a parent most of the time, so give your new relationship all the time it can get to just 'be' without kids.
Your Ex (our OP) is NOT being petty, she is naturally concerned and in this instance, i think she has a right to.
FWIW, I was introduced to my OH DC way earlier than we'd planned, but the ex wife got wind of his new relationship in the early days (like date number 3 or 4) and filled his DC full of so much crap it made them want to meet me and my DC. She was trying to scupper our relationship, or scupper the one between her DC and the dad. It backfired, we get on like a house on fire and I really value the time we spend together. We both do however vigorously work to ensure we have relationship time together and time with our own kids.
It's a balancing act and can't please everyone all at once, but the key people in all this is the DC. Don't rush them, it's not fair on them.