How to tag a desire path? Should they exist at all?

We map what is there, not what should be there. You should tag the legal access as additional information, though.

So, for https://en.wikipedia.org/wiki/File:Desire_line.jpg I would go for

• highway=path
• surface=ground
• smoothness=bad (just a guess from the image)
• width=0.4 (if I would have measured/guessed it from reality)
• if needed: access tags depending on the legal situation and signage. If a sign forbids use officially (whyever it is there) but no one (including the owner) cares, then I possibly would tag it access=permissive + access:legal=private (or something similar … I am not sure if we have an established tag here) + access:description=officially forbidden by a sign which no one including the owner cares about (a tag describing the access to make it clear to users and mappers). Similarly, if a sign forbids use by all vehicles, but clearly only motor vehicles are meant (just the wrong sign was put up) I would not use vehicle=no. Note the suggested default access assumptions.

Somewhat questionable is a "desire path" which crosses a railway line. Is there a path for the 5 metres across the railway line? Likely you cannot see it, you can just assume some use, but there is no real-world feature. In this special case (also influenced by the danger) I would tend to not map the part which is across the railway line.