I think the source of the problem you are having in understanding the potential of Dick's result is that you are not familiar with differential equations. Differential equations, in general, are notoriously difficult to solve. Moreover, unlike algebraic equations which I'm sure you are familiar with, solutions to differential equations are not numbers. Solutions to differential equations are functions. And, in many cases those functions are not simple either.
You asked for an example, so to illustrate the power and the difficulty, consider Schroedinger's Equation. It, too, is a differential equation and is difficult to solve. But, when people figure out solutions to it, it yields incredible insight into reality that can then be exploited to make things like lasers and Bose-Einstein Condensates.
Dick's equation is even more general. In fact Schroedinger's Equation is just one of the solutions to Dick's equation. So when I say that Dick's result can provide a new avenue of approach to science, I mean that people can begin to look for new solutions to his equation. There is every reason to believe that if a new solution were to be found, it could be every bit as fruitful as Schroedinger's, or Maxwell's, or Einstein's equations, all of which are solutions to Dick's.
Now you object, well why did Dick only pull out the already-known equations. I tried to tell you before, but I'll try again. Compared to finding a solution to a diff. eq., checking a candidate solution is vastly simpler. In other words, it's relatively easy to take Schroedinger's Equation and test to see if it is a solution to Dick's equation, but by comparison, it would be very difficult to find it with no previous clues.
I tried to explain this to you once by comparing the difficulty of finding a factor of a large integer with the difficulty of testing a specific integer to see if it is a factor. To test it, all you need to do is divide and see if the remainder is zero. A grade school level task. But to find a factor of a large number is a task that is generally beyond human capability and even beyond the capability of most computers in a reasonable period of time.
Now, what part of that did you not understand?