Sure, but I'm not sure how that is related. What I'm essentially talking about is you have a DE that is of the form -i h\phi_t -b\phi_{xx} + V(x)\phi = 0 and graphing \phi as you vary V(x). That is not obvious unless you know how solutions of the equation behave for different simple cases of V(x) and continuity. That, I argue, is intuition.

Actually, what you mentioned sounds like intuition to me. You didn't make hand plot the polynomial, so that is relying on intuition over rigour, which I think is what the OP's quote from Feynman referred to.