(I’ve now also read the proposal text. I wanted to post this in the discussion page but read that you prefer to have feedback here. I feel discussing it in the forum is usually less focused and productive due to the format. (No threads in the forum, no issue-oriented feedback, i.e. per-topic), but anyway, as you wish)
The current proposal text already has my vote, but here are some comments that may lead to more people approving of the proposal:
1. long proposal text for small change
The proposal reads a bit long-winded and complicated. It is just about recommending to use cycleway:both when users want to explicitly specify that indeed both sides are the same, yes?
I’d recommend at least cutting the Looking at numbers section short, just add a graphic from taghistory (and annotate it with some labels, if necessary).
2. tag definition
From the discussion in this topic so far, I take that some strongly disagree with the notion that cycleway means
cycleway=*- The given bicycle infrastructure is present somewhere on the road. It might be the left, right or both sides.
(emphasis mine)
…because they have been using this tag for decades and might even feel like this text might be like a redefinition. I’d recommend softening this wording, degrading cycleway to something alike highway=road (as I’ve read having been mentioned in this thread) seems to overshoot the target.
It might be enough to mention cycleway is not explicit about for which side the cycleway is meant, but usually it can be assumed that it is meant for both sides except in the case I mentioned.
3. editor support
The greatest part of what you would like to solve with this proposal seems to be (approval for) a new feature request for iD - to properly support cycleway:<side> alongside cycleway and not just hide one of these tags if the other is present. TBH, that seems almost like a bug in iD. Maybe what is needed is rather someone to properly implement that.
(OK but I understand that the issue remains what such implementation should tag in case both sides are specified to be the same - cycleway:both or cycleway. At least one person in this thread seemed to not have been happy about that StreetComplete tags the sides explicitly if the user explicitly specified each side in the UI)