Why don't high-energy experimentalists ever include Faddeev-Popov ghosts in their Feynman diagrams?

OP is asking how can phenomenologists get away with neglecting the FP ghosts. The answer is that they can't. In a talk you can omit irrelevant stuff to keep things simple. In the full computation, you must include the FP ghosts (or use a ghost-free formalism, which in general is much more cumbersome; e.g., the axial or unitary gauge).

For an explicit example of a paper from this week (from the phenomenology section on arXiv), see Dynamical Symmetry Breaking by SU(2) Gauge Bosons. In particular, in the appendix you will find an explicit computation that requires ghosts (although the relevant diagrams are not shown, but referred to an older paper). See also The method of global R* and its applications. For an example from last month, see Evidence of ghost suppression in gluon mass dynamics. There are countless examples. It couldn't be otherwise: you do need ghosts to have gauge-invariant results. If you did not, why would people introduce them at all?

Perhaps OP does not usually find ghost loops in phenomenological papers because the latter tend to use results from other papers instead of calculating them themselves. For example, the beta function of a QFT is an indispensable object that is used all the time in phenomenological papers; but these papers tend to quote the formula from theoretical papers where it was first computed. No need to compute it again. Needless to say, when a theoretician calculated it, they did use ghosts. The phenomenologists simply shows the result, so you won't see the ghosts there. But, like the real ghosts, they are there, whether you see them or not.

Alternatively, another reason is that many phenomenological papers are usually more concerned about the qualitative description of the system than the quantitative details. Therefore, tree-level computations usually suffice. And as ghosts only appear in loops, they are irrelevant for the tree-level result, and may therefore be neglected. But as soon as you want to include loops, make sure to bring them back, or otherwise your calculations will be wrong.

Perturbative QCD computations can be done in ghost-free gauges, such as the axial gauge. As far as I understand, this is customary for the computations revolving around the parton density functions in the proton, or in the pion. As well as for the final state equivalent, the parton fragmentation functions. Combined, this makes for a rather large section of QCD phenomenology I think. This might be one reason explaining your experience.