Sure, but all of these things are always interlinked. It's entirely possible that the first deviation we ever find from general relativity will come in the form

Sure, but all of these things are always interlinked. It's entirely possible that the first deviation we ever find from general relativity will come in the form of just trying to make better and better clocks, but then finding some funny systematic effect that prevents different atomic clocks from properly syncing up, and then ruling out all other explanations. This kind of discovery is not without precedent in physics.