I am a father of 15 week baby, and taking an enormous number of pictures every day. "From new born to toddler," I think you do not need good AF tracking (he is not moving much yet), long focal length (we stay close, right?), and very fast aperture.
However, F2.0 is often needed for indoor photos. Faster than that for me personally has too shallow depth of field.
In this sense, I think X100T is the best among fixed lens X's.
You can probably start to think of other needs one year later.
By the way, I am using X-E1 + 35mm F1.4 indoor and + 27mm F2.8 outdoor.