I wrote “purpose-built shared cycle track” because that’s what I meant. If you take the Bristol and Bath Cycle Path, or many other cycle tracks on former railway alignments, they were built with Sustrans funding as unsegregated shared cycle tracks by design. Tagging them as anything other than highway=cycleway would be absurd.
Then leave them as they are. A way in the UK tagged as highway=cycleway has implicit foot=yes access unless tagged otherwise. Guidance on pedestrian priority is irrelevant, as highways where cyclists and pedestrians are allowed but pedestrians do not have priority do not exist in the UK.
I agree. Adding cycleway=sidewalk and tagging width=* or est_width=* would be useful in order to identify low-quality cycle infrastructure created by little more than adding signs. Changing them to highway=footway because of a misunderstanding of how pedestrian priority should be mapped (it shouldn’t, unless it differs from the default) may not be quite as helpful.
Before you, as a relatively new user, unilaterally decide that everyone else has been wrong for years, maybe it would be appropriate to discuss it first?