Where would they move to? And if you say to whY class then perhaps you can explain why they should tolerate a crying baby where hundreds of pax will be disturbed compared to the 12-16 in F?
I have travelled in all three classes and know there are enough noisy people in all of them. I don't want problems from another cabin being moved to mine just because they can be and that as a lower class pax I am perceived to be of lesser importance and have to tolerate what others don't want.
I think the option to request passengers with a crying baby move out of the First cabin is about passenger expectations. If I book a ticket in WT or WT+, I know that one of the non-financial costs will be that these are cabins full of children and there will be crying babies in them. Indeed the BA's description of the WT cabin focusses on the children there, so you know that is what you are getting. I would not complain about crying babies if I were travelling in those cabins.
The description of the First cabin is completely different and so are the passenger expectations for a quieter calmer environment. It is not about the number of passengers disrupted by a crying baby, but the intended environment of the different cabins.
If First passengers find they are exposed to crying babies their expectations from First will not be met and they may feel there is insufficient additional benefit from the cost premium to travel First and switch to CW travel or to the WT+ cabin.
If BA could guarantee the First cabin is child free, demand for travel in First could increase as passengers know the cost premium from CW is worthwhile to provide a quiet cabin environment. After all, there are more people travelling without babies than who travel with babies.