The relation role ‘alternative’ is meant for official (and usually marked) alternatives for an official (and usually marked) hiking route. They are not good for unofficial shortcuts or desire/social paths. These should never be added to a hiking route relation.
There is a tag informal=yes (to be added to the highway=path way) for desire paths which is favoured by the US trails access project (see in particular the section ‘Suggested Tagging’). But this is all still in a state of discussion.
Personally, I would suggest to not map these shortcuts at all. If they are there as highway=path, they will be used and routed over.