Is it possible to intuitively understand hybridization?

I am circling the drain trying to grasp this. I can somewhat understand why hybridization occurs with sigma bonds and to a lesser degree lone pairs. What I simply cannot understand is why pi bonding electrons are left out.

The explanation I keep coming across is that the pi bonding electrons do not influence the shape of the molecule and therefore do not hybridize. This to me seems like an effect of the pi orbitals not hybridizing and not an explanation.

Am I searching for an intuition that does not exist? I understand this is a model and as such it has limits, but I would like a better intuition for how these things work.

Has anyone developed a comfortable intuition for these concepts? If so, what was the breakthrough?

I think of Feynman's quote "nobody understands quantum mechanics" and wonder if I am encountering what he was describing...