Hang on.
Your children are too small to be at the stage where you 'do family stuff' in an easy, let's have fun, let's go out for a Saturday afternoon kind of way.
They are tiny. One is practically a baby at 2. Time spent with them isn't going to be just fun family times, it's hard core parenting. Feeding, changing, taking to loo, holding hands, doing certain things at certain times.
They are also at the stage where yes, they really need you to be around and present all the time as the parent in the home, when they are in your home and not with their mum.
Ok, you say you don't expect her to do the parenting, but the fact is that just by joining you in day to day stuff when they are there, she has to spend her time parenting by default, because that's what you're doing, and it's kind of exclusive to doing anything else at their ages. And I think when you moved in together you simply assumed that that's what would happen. But that's actually a massive ask. There's a reason that people - usually mums- with young children usually hang out with other mums with kids the same age. You HAVE to do certain things and it's quite difficult to facilitate much else and it's boring and frustrating unless one of said kids is your precious wee thing who you find fascinating. And even then...
You split from your children's mum pretty quickly, possibly even before the youngest was born? I'll be honest, it's great that you're stepping up to parent but I think you honestly expected that simply by starting a new relationship with another female, you would have an adult who would automatically simply fill that 'other parent' gap and just be happy to plop her life into the category assigned for her.
Good that she isn't. Good that she's making space for YOU to parent, alone, first and foremost, so that your tiny children really bond with you, the parent - as they are supposed to. At this age, it would actually be quite shitty parenting if they were always spending that contact time equally with another random adult who isn't their parent. This time is their time with YOU.
As for the sleeping etc... well, this is where you probably both rushed into it. Honestly... it would have been better for you to say 'I'm a dad, right now my life is pretty intensive parenting, that takes priority, maybe living together isn't right yet.' Back to my first point though. I think you thought you were getting a parenting assistant as well as a partner, but - you didn't. And that's fine and has a lot of positives for your children.
I think she's perfectly happy, likes your kids, is happy to share a complex life where you've both got lots of very DIFFERENT commitments, your main one being parenting YOUR children. But she hasn't signed up to co-parenting them, no, and there's no reason why she should.