You can do non-perturbative QCD with diagrams. You just need infinitely many of them. :p That's, by the way, the only way I know how to actually do computations of proton structure. Approximations are made, and corners are cut, but it's not bad, all things considered.
But more importantly, we shouldn't be tailoring our jargon for the laymen. Scientists and medical doctors used to speak Latin specifically to exclude the laymen. Now, the fields have gotten complex enough for it not to be strictly necessary. (Though, I think we should have stuck with it.) It's very important to be able to quickly distinguish between scientists and people who try to pretend to be scientists.
When a student doing field theory asks about photon exchange in a scattering process, I can immediately tell a difference from someone who has no idea what they are talking about. In the later case, I'm going to try and lead the discussion towards a classical case, if it is possible. There is absolutely no reason to try and explain details of how QFT works to a person who doesn't know anything about QFT.
OP has been asking questions which amount to asking if we can pin-point a specific quark in a proton. Answer is no. And I have no better means of explaining it than saying that quarks are constantly created and destroyed. You think you can explain the full case, where number of quarks is a distribution to this person? You give it a shot.