They could, but it'd take a lot longer and movers are usually paid by the hour so they could expect to hear a lot of "Stop playing on your phone and get to work" or even "why are you taking photos of my (or my child's) stuff" which could get pretty awkward.

I expect that right now moving companies do sell info like your name and your old/new address. That alone should give them a rough idea of your income level, if you're getting poorer or richer, starting a family or recently divorced, etc. If you're using a moving company like Flat Rate and taking photos of all your stuff so that they can make an estimate they could be selling that data too.