From the picture it seems that the green path does not go through a mapped pathway.

OSM won't tell you to go over random terrain, it will only guide you through paths that have been mapped as walkable. For all it knows, that terrain is thick bushes or something.

Because there's no routable path there, and if there's no indication of such a path being there, it should not be mapped. There's cases of informal paths @roux2scour@jlai.lu's picture describes, and may be tagged with "informal=yes". Other than that, I'm not aware of any routing services, which would behave like you'd want them to. The path of the starting point also seems raised (as being on a hill), compared to the endpoint; so I'm not sure how desirable this particular case would be, even if it were to be supported.

it is telling you not to step on grass

Why would you expect routing software to route somewhere that no path is mapped?

As for public transport routing, I explained this here https://discuss.tchncs.de/post/50719308/22594280 a while ago.