Not at all, I don’t want to leave anyone with that impression! There is no general right to roam, and I quite often add access=private to rural tracks and paths.
My point was that where paths are open to the public, as in the two examples of public land I mentioned, there is often no distinction between pedestrians, horses, and bicycles. It is “motor vehicles” that are usually subject to prohibitions. If a land manager wants to permit walkers but not cyclists (or horses) they would need to make that visible in some way, as they do in the nature reserve I mentioned.