Maybe (as @multimodaal s and @Minh_Nguyen point out) a new key might be needed to make this work. But it’s a really complex matter. Very good points so far, let’s keep them coming and the discussion alive!
The point on =steps, is a good one for example. Here, step_count= matters as well. I might cycle through (or carry my bicycle on) two or even three steps if it provides a substantial shortcut, but dismounting and carrying a heavy bicycle over many steps would have to save multiple kilometers to be even remotely worth the while.
This is a good point too. And the access tags aren’t really meant to describe physical accessibility (more on this just below), right? I’d imagine it very difficult to come up with good or even somewhat objective criteria particularly for suitability/practicality. Again: enough descriptive tags might be more helpful and less controversial than a new key…
Indeed! One way I’ve seen access=no being (mis?)used is to tag ways that are under construction (or undergoing slight renovation, but long-term, and fenced-off). It seems that recently some routers began dismissing the ⟨transport⟩=no tags on ways that still had a ⟨transport⟩=designated tag (i.e. route through them even though they had an access=no tag). I suppose that is the way to go. I can, however, also understand the argument that if the way (e.g. a cycleway) is still substantially unscathed by the construction and the traffic signs are still present—but the ways are fenced off for construction—they are in some sense still designated even though not accessible to anyone.
The point @Carnildo made about skiing is one I’ve encountered also. A sizeable portion of paths near Helsinki (and other Finnish cities) get turned into ski tracks in winters that have enough snowfall. Even though walking or cycling on the prepared ski tracks isn’t really illegal, it is frowned upon in the extreme. As in risking-your-front-teeth-getting-knocked-out extremely frowned upon. Since the ski tracks are prepared only in winters with sufficient snowfall, these paths are not usually a part of piste-relations indicating that they might be unusable during (some) winters. This would be a suitability category too (in that it is not captured with descriptive tags nor a legal restriction)?