I would think it's expected, as global, container and summary fields are omitted from quickfinds.
Assumably then any relationship based on an omitted field is also made invalid.
You might want to try and calculate a field with List(relatedTable::firstName) and add that field for quickfind. Not sure if that would work or not.
The behaviour is correct and consistent, since for a relationship that works via a global, the related name field will be the same for all records; therefore, if QuickFind is successful, all records will be found (what you interpreted as failure); but if isn't, the found set doesn't change (same behaviour as with a regular Find).
Let's just say that this way of searching doesn't really make sense; it's more meaningful to search in related records where the relationship uses a record attribute (field data), rather than a global.
Your explanation makes sense. I will now have to disable the quick find for all fields on my layout that are related via a global.