My only “intuition” (which maybe isn’t right enough) is: if the “speed of the photon is constant” then other things we’d believe to be constant just aren’t. Which in my imagination (of course influenced by what I’ve read up to now) results in weird deformations of spacetime happening all the time. Adding the gravitational fields which we intuitively feel similar to the acceleration (e.g. falling down makes us weightless) makes the deformations even weirder. So my “intuition” is “the spacetime always deforms to make everything as we finally observe.” Nothing “picks” anything, the stuff does what it does in the already deformed spacetime, and by doing what it does it can also deform it more or less while it does what it does, and everything fits at the end. The only thing that’s fixed is the speed of the photons. Weird, but at the end, only the formulas can allow us to calculate what we can expect. It’s simply that on the scales of the “ordinary” stuff for us the things aren’t recognizable, that’s why we don’t have to have the intuition about that. And why most of the things we observe without some clever instruments (or without using the clever formulas) are “intuitively normal” for us. According to the story even those who planned the pre-GPS satellite made the hardware to turn on or off the compensation of the effects of the relativity, just in case. But the measurements matched the relativity to 1% accuracy. Not everybody is Einstein, including (obviously) us, for we wouldn’t be here discussing.