The best thing about "your speedy"

I love my speedy. It's truly a bottomless pit. No such thing as a speedy that can't carry something if you buy the right size speedy. For your needs I would look at the 35 or 40. Don't rule out the 30, but the 35 will most likely be the one that grabs you. I would also get the bandouliere version because you will probably need to be hands free. You can always take the strap completely off for a night out. The speedy is an iconic bag, and will stay that way forever, as it already has. It can hold all your needs if you buy the right size, and the crossbody option makes it super comfy.

There are plenty of other LV bags that are perfect for mommies that aren't the neverfull or speedy. Some off the top of my head are the totally, delightful, portobello. LV has many hobo style bags and tote style bags that carry A LOT.
 
I just bought a Speedy 30 Bandouliere in Damier Ebene last month. I did a thorough research on it, I finally made my decision because I found a "made in France" bag!!!

Pros: very Classic, relatively inexpensive, very light, very versatile

Cons: everyone has a Speedy, and this bag is very easy to counterfeit