Well I think we should ignore the exceptions and not make people abuse them without a reason. Bicycle=yes can be added on footways where there is no good alternative, but that's besides the point.

In this case there is a crossing where both sides don't have bicycle=designated or anything about bicycle access, but the crossing itself suddenly has it. IMO it's a bit strange that crossing has more access than the paths it connects.

As for iD tooltips, I don't think we should 100% trust them. I don't mean that there is any big problem with how the things are mapped, but maybe in future we can be better.

One other thing I'd like to also get rid of is a 

highway=footway
bicycle=designated

at least on non-crossings, as this is not the best combination and you won't really find wiki examples of it. Some clients like Komoot treat it differently.

If the path is signed, then a different combination is better

highway=path
bicycle=designated
foot=designated

(or at least the iD-way of highway=cycleway, this time CyclOSM will be confused)

Sometimes there is also footway=sidewalk, but as it turns out 

highway=path
bicycle=designated
foot=designated
footway=sidewalk

is perfectly ok
https://wiki.openstreetmap.org/wiki/Tag:footway%3Dsidewalk