eInk does seem like an awesome choice for this to me too.

I've gone without a bike computer for the same reasons as not wanting a second device. I used my phone for years in a water proof front frame bag with a transparent cover. Usually for not more than 3 hours at a time. Only on my Pixel 7a did I end up with burn in from the Wahoo Android app's status screen, where "n/a" for not having a power meter ultimately burnt in after about 2 years, at least that's when I noticed it. Now I just go without being able to glance down and see ride stats or directions. More planning or stopping if I reroute is required, but I didn't want to damage the phone... :(

Presumably an app could cycle displayed screens, if that's what you meant by switching modes, to avoid burn in? But I really thought AMOLED was past that when I never saw it on previous phones.