I'm not saying it's more or less valid than anything else. I'm saying it offers insight as to what people within the bicycle industry are willing/able to test, and in particular the lack thereof.

No, they definitely do test. I'm saying that the testing probably isn't as broad and holistic as you think.

For instance, I would expect that it's rare or nonexistent for racing teams to have a bunch of otherwise-equal copies of a frame made with varying stiffnesse in order to quantify the impacts. That's in the realm of things that are messy and where you can't tweak things for rapid turnaround. And that's the sort of thing that would have to be done to make a useful judgement on the effect of frame stiffness.

Yes.

Stiffness should matter to how a frame behaves as suspension. Plenty of modern frames try to smooth the ride by allowing considerable vertical flex to occur, sometimes adding an explicit damping mechanism to prevent ringing.
That's one area where CF may have an advantage over the metals; it seems like it tends to be a more imperfect spring on its own. Titaniums and steels can often be allowed to flex by a huge amount without durability concerns, although being very undamped, that could cause excessive rebound in some circumstances.

As far as pedaling...
I recently changed the cranks on my gravel bike to test how it would feel if I made the pedal position closely match my road bikes. Not a big change, but slightly narrower Q and slightly shorter arms. However, the new cranks are a bit on the cheap side and doubtless much stiffer than the old ones; both use solid aluminum arms, but the old ones are decent spindly vintage arms while the new ones are very fat. Since that change, which I will soon revert, it's been nigh-impossible for me to stress my cardio on that bike; I simply can't get my lungs to work hard even at very high RPMs, except while climbing. The speeds I can maintain on flat ground are noticeably slower, and my quads fatigue extremely quickly if I try to go fast. It is exactly in line with the sorts of differences described in the results of the BQ frame flex study for when someone's legs weren't liking the characteristics of the bike.
I find the argument that flex about the bottom bracket can be significant in terms of the biomechanical side of the pedal stroke to be pretty convincing, even if the mechanism isn't well-understood.

Hard to say. I'd say that needs further study. Intuitively it seems that increased rider power should play nicely with increased stiffness, although without the effects of frame stiffness being characterized across demographics, and without the mechanisms behind its effects being well-understood, I hesitate to make a particular judgement. Sean Kelly seems to have enjoyed riding a screw-and-glue wet noodle, at any rate.