LOL I could have written you post a few years back. I think that I went through at least 5 bags before finally found the one I really like.
I wouldn't get a Neverfull. You will have to fish through the bag to get what you want and everything will spill out should it fall. A couple of questions:
-do you bf or ff? If bf then you really don't need to bring bottles so then you will have more options.
-is this bag meant for trips longer than a couple of hours or all day? If just a couple of hours, then you get away with just a tote with a zipper compartment. If it's for the entire day or longer than half a day, then I would get the handbag you really want and store the kids' stuff in something else. I always like you ordinary backpack. Lots of compartments to store their snacks, drinks, diapers, wipes, extra clothing, etc. And it is gender neutral as you can get your dh/dp to carry it for you.
-is your toddler into carrying his/her stuff in a cute backpack? If this is the case then you may not need a big bag.
My favorite was the Nouvedive leather messenger bag with a zipper compartment & several small pouches inside. It wasn't a LV or
Gucci or any famous brand but it was great as it didn't look like a diaper bag.
HTH