That's great idea! Standardizing on an interface that can be implemented by standard phones as well means there'd probably be much better and more widespread support.
I suppose they could even support the "small secondary screen" interface on phones without a physical secondary screen, by simply displaying its contents inside the lock screen.
I used to have a foldable phone with a small secondary screen like that as well; it really was nice, and would be even better if it didn't need a button-press to activate (e-ink or whatever), as turning on the screen is always about 80% of the effort when I just want to see the time... [they can't make the button too easy to press after all, to avoid accidental activations...]
