Already reached the conclusion when I saw a genetic researcher presenting his p-value < 10^-40 as better than < 10^-10. I kept my mouth shut because I didn't want to ruin the poor guy's moment in the sun, but I knew it was time to get out.
I know there's always more nuance in statistical reasoning, but the first number is vastly smaller than the second one, right? Is it just that both are hilariously tiny and not credible? Or is there no additional value after you get into the one-in-billions territory?
A nice demo of this effect is DNA matching in criminology. Although DNA matching of suspects to DNA samples can be insanely accurate, in practice it is limited by the incidence of monozygotic (identical) twins, which is about 3 in 1,000. You cannot be more certain than this that you got a match, essentially.
This example is a gnome-wide genetic association study. Every genetic variations are tested, so at least 500K or more linear regressions were performed. This many statistical tests could lead to many false positives just by chance, so one must do multiple-testing corrections. The end result of multiple-testing correction is much bigger and therefore worse p-values. Hence the drive toward ridiculously tiny p-values.
Some tiny aspect of the real process that your model falls to capture might mean that that 10^-10 is actually 0.001, and 10^-40 is also 0.001. In complex biological fields it's fair to assume that there are always such tiny aspects.
I think you're making the classic Prosecutor's fallacy: https://en.wikipedia.org/wiki/Prosecutor%27s_fallacy. In my experience, smaller p-value tends to be more of a measure of sample size than anything else, or an overly restrictive null distribution that is almost certain to be rejected.