The comparison to children playing with blocks makes me wonder: If you spent long enough playing with these 4D toys, would you develop a deeper intuition for the 4th dimensional shape of the objects?
Interestingly, I was told one of George Boole's daughters got lessons from a guy on 4D objects by using colored wooden blocks. She really picked it up, and later in life mathematicians would consult her about intersections of 4D polytopes.
I’m sure most of us could learn a visceral understanding of 4D if we could push and twist 4D objects.