Big questions like that are always interesting to tease out -- what if there were no more Z bosons?
1) First off, that would mean the math / symmetries of the standard model would break. The way the laws work, you *need* a Z boson (just like we *needed* a Higgs boson, and later found it). So, in some sense, you couldn't just get rid of it.
But that's so unsatisfying!
2) So let's imagine that, however we did it, we managed to destroy the existence of Z bosons forever. There are a lot of particle interactions that the Z boson takes part in, and they'd all be affected. To take a simple example, when two charges repel, there's a photon being exchanged. *BUT* it could also be a Z boson, standing in for a photon. (It has a far lower probability, since they have such a large mass, but it's non-zero). I'm not sure how that would all trickle down, but knowing how fine-tuned the knobs of the universe seem to be -- any small change could have big repercussions.
3) As an even more concrete answer, though, there are some things that *only* a Z boson can do. Classic example: elastic scattering of neutrinos. Neutrinos only interact with the weak force (hence only with W and Z bosons). If a W is involved, since it has charge, some *other* charged particle needs to be involved, too. Perhaps an electron is turning into a W- and an electron neutrino. To have elastic scattering, though, where neutrinos are simply changing direction, otherwise unaffected, you need a Z boson.
So there you go, neutrinos would travel in even straighter lines than they already do
That neutrino scattering was actually something I spent time working with as an undergrad. In neutrino detectors you're looking for both charged currents (W is involved) and neutral currents (Z is involved). We had to go through images of events and pick out what was happening in various particle showers. So, what I'm saying is... if there were no Z bosons... that undergrad project would have been easier.
Huh... suddenly I'm on board with this idea.