You've stumbled on something quite important. The approach were we try to model the type of structure you are hinting at, is called "multilevel modelling" (also: "mixed-effect model" and a bunch of other confusing names).

I'm not really sure whether this approach offers advantages when trials are truly identical. But it seems to me that they rarely are (maybe on a very simple task, say where you distinguish "X" from "O" in an "oddball" task). More typically, they are assumed to be some random sample from a population of possible trials. For instance, I used to do some psycholinguistics stuff a long time, and there we often had words with a certain property. There, it may be very useful to model your item as a random factor.

Another thing to consider is that even with identical trials, there are generally training and/or fatigue effects over time, and that these differ by subject. By modelling these, we can reduce unexplained variance and increase power.

Here is a paper on this topic within the domain of psycholinguistics. It assumes items as a random factor, so it's not a perfect match, but it helps to explain the concepts.