Do Traits Have to be Adaptive in Order to Survive?
Good question. And good analysis. I have little to add! I'll simply provide my own list of thoughts to complement your ideas, which are not mutually exclusive.


  The fact that it wasn't discarded during the course of the species'
  evolution suggests it must have offered some benefit.


This statement is speculative. The key word here is suggests. i can think of several speculative scenarios, of course the list is by no means exhaustive:


Traits that are not adaptive can still exist in the organism as vestigial traits. This can be because the trait was recently useful but no more, over evolutionary time scales, and not enough time has passed for the trait to be lost.
The trait may also be floating, i.e. has not been selected against, and acts as a neutral trait. It therefore remains as a phenotype. Traits are theoretically unlikely to float for very long periods of time, as I will describe later.
Some traits can very indirectly affect other traits. In practice it is often difficult to completely describe or understand which traits are a result, or cause, of other traits.
Many traits can also be byproducts of other useful traits. With nipples it might have something to do with early exposure to hormones; developing a way to bypass male nipple formation could be difficult and costly. With psylocibin, it may be a side-effect of a particularly useful biochemistry to the fungus, perhaps metabolic, that is completely unrelated to its neuroactivity in humans. It may be that a biochemical pathway might form psychoactive intermediates or products that are difficult to remove or metabolize, and find themselves in the mushroom co-incidentally.
The trait may simply be very useful, but we don't understand why it is useful. Humans often overlook this possibility; for instance, I assume you have not considered that psylocibin may have unknown effects on non-human or even non-vertebrate life.
I am not an expert but filamentous fungi sometimes/often have very complicated sexual reproductive strategies. Psylocibin may be maintained through sexual selection, rather than natural selection. This is probably far fetched, but hey- we're speculating about our ignorance.
Lastly, but very importantly, all organisms face pressure to survive and replicate. It is expensive to maintain useless features and traits, or to produce excesses, or to overdevelop or harbor chemistry that is not doing anything. Natural selection is ever-active in limiting in which traits are maintained within a population. This I think is the most convincing argument; the principle of economy. This argument fares well in all discussions on biology; life is difficult, and the costs of survival are very important to consider, always. It is expensive to store or maintain constituent chemicals, even if they are not toxic, nor if they have no use for the survival of fungi. It is more economical, and less costly, to lose such traits; selection will favor fungi that save or spend this energy elsewhere. This is a very important point, and I think it is the most salient to understand the quote you provide.


I think it would also be a good idea to study the number of times psylocibin arose. If it occurred ('was invented') multiple times independently, it must have had some functionality for the organism, like wings, or vision. If it occurred only once, one can ask whether the majority of descendant strains has lost it, or whether they maintain the trait. That could also shed some light on whether it is maintained through positive selection. Lastly, psylocibin may be a chemical that allows to explore a very specialist niche of the ecosystem. If few descendant strains contain it, that can mean it is still selected for, though only by fungi that rely on it as a strategy. This would also mean that psilocybin-less fungi would have found no need to maintain it, and may point to the occurrence of negative selection. Hope that makes sense.

Mandatory reading for you on relaxed selection.

tl;dr yes, psylocibin may be maintained by happenstance, though likelier via positive selection, at least in specialist fungi