It is true that a noun in the partitive case cannot be a subject. The special у/ю declension applies to a small class of masculine nouns, and either a noun or a verb may establish the case. As partitive is mainly used for mass nouns, or variable quantities, the case ending can be seen, in context, to denote "some" amount of an item or substance.

Furthermore, partitive case endings are to be used with perfective verbs.
Выпить чаю: proper. "To drink some tea". People may use the other variant, but it is not fully grammatically sound.

As for nouns preceding partitive case, the unique ending relates to parts of a whole. Imagine taking a cuplike amount of tea from a larger supply. In communicating this action, чаю would be used, emphasizing a definiteness: чашку чаю, "cup of (the/that) tea". Even so, outside of verb usage, the traditional genitive case ending for partitive nouns is more universally accepted.