"Grammatically Animate" = "having the Accusative Case the same as the Genitive" (opposed to "having the Accusative Case the same as the Nominative", which is inanimate).
This definition has nothing to do with the lexical meaning of a noun. For instance, мертвец is animate, so we should say "Я вижу мертвеца", not "Я вижу мертвец".

Animacy is applicable to all the plural nouns ("вижу книги" vs "вижу кукол", "вижу столы" vs "вижу детей", "вижу окна" vs "вижу существ") and for masculine singular nouns having zero ending in the Nominative ("вижу стол" vs "вижу мертвеца").

Indeclinable nouns still can be animate or inanimate, which is reflected in the Grammatical Case of an adjective:
Animate: "Я вижу больших пони"
Inanimate: "Я вижу большие кашпо"

Feminine and neuter nouns having no plural are neither animate nor inanimate (пыль, посуда, зверьё and so on).

What about начальство, человечество, студенчество, they do have plural, though their usage in plural is rather rare.
So we can check their animacy. My intuition of a native speaker tells me (and a dictionary confirms) I should say:
"Я вижу начальства", not "Я вижу начальств";
"Я вижу человечества", not "Я вижу человечеств";
"Я вижу студенчества", not "Я вижу студенчеств".

Hence, they are inanimate.