Has the tag become so broad as to be (relatively) worthless?
Personally, I think it's more an issue of ignorance on the asker's part that many of the Questions here get flagged with the mechanical-engineering tag.
Being trained in engineering, it's easier for a lot of us to distinguish that certain topics should be tagged certain ways. For example, a question on beam bending for a building system could be tagged with mechanical-engineering and that wouldn't be technically wrong. However, I think structural-engineering or mechanics or statics are more fitting since the question is about a specific topic of mechanical engineering.
Eliminating the tag would only serve to confuse. For example, a question about designing an engine based on a set of criteria could have the engine tag, but the mechanical-engineering tag is still the most appropriate.
Does some thought need to be put into describing the limits of the tag?
Looking at the tag's entry, it seems pretty well-defined. It even says
However in order to use this tag the question should be about a mechanical system or machinery.
At this point, it's just the user not reading the wiki entry before adding the tag, which is understandable for someone new to the site.