(Too long for a comment and it is kind of a soft question anyways)
I'm not so sure that your assumption of an imbalance is valid. Maybe it is. But would be better if demonstrated (or at least explored) first. Otherwise, we end up finding an explanation for a phenomenon that doesn't actually exist. Plus the exploration would probably inform the answer, ...
Analysis is useful
Physics uses analysis. Engineering uses analysis.
Continuous models are widespread everywhere, and typically they look like differential equations, if there is time involved.
Finance uses lots of analysis.
Note that a lot of modern geometry is differential geometry, which builds on a foundation of analysis.
Certainly, other parts of ...
I can only help with (3).
A. This behavior is not unusual and not just with intuitionists. It's good to be able to do things in your head, but you need to "know your head" and when you will have issues.
B. In general, when doing pen and paper you should try to write down all the steps prone to an error. Of course there's a balancing point. But ...