I have always thought the eyes were for the fisherman. Back when I started tying flies JC was about impossible to get and the plastic substitute was kinda hard to work with and harder to look at, so not knowing that they were representing fish eyes and ultra important, I tied a lot of flies that called for JC eyes without them and every one of those caught fish.
I have since seen methods of using a similar shaped feathers and painting the spots on it, and the plastic now are much better, and now the real thing is more available, but I'll continue to skip it.
Trout that eat sticks and pebbles that only slightly resemble foods may be stupid enough to overlook the lack of JC.

At least now you can do a side by side comparison. I still have some but stopped bothering with it long ago, my experience being about like Trev's. But what fun to get a buddy and fish identical flies except for jungle cock "eyes" in the same water with the same presentation.

If you experiment long enough, you might find a situation where some form of "eye" or, essentially, gill and pectoral fin coloration and motion make a difference. Jungle cock--an exotic feather with an exotic name that makes for good fly fishing mystique--is just one schtick in a realm that has expanded greatly with synthetics, marking pens, and so on.

What a fly looks like to a fisherman is very different from what it acts like to a fish. So deciding which you are tying for and choosing materials accordingly, focusing on presentation rather than static appearance, is most important.

But a book of streamers tied with jungle cock is a wonderful thing, and I'd be very surprised if a side by side comparison ever showed they did not produce as well--unless they were tied in such a way as to limit motion of other forward elements of the fly.

Personally I think the JC eyes don't mean much to the fish. But that said, the flies tied with them are much prettier to the angler and in turn the angler fished them with more confidence and then takes his or her share of fish.

Have experimented with it both ways with JC on a favorite pattern for trout. It does matter.

Many eye substitutes available. For most species, find adding eyes in some form beneficial for streamer patterns.