Semantically, none is neither singular nor plural. It's less than one and much less than many.
So its subject agreement is entirely arbitrary. Plus, negatives are noted for their funny grammar.
Positive quantifiers are either singular in verb agreement, like each and every, or plural, like all. None can be either, depending on context.
If it quantifies a plural noun phrase, as in none of us, then it's equivalent to the negation of a universal quantifier in many cases.
- None of us are going to the party.
is equivalent to
- All of us are not going to the party.
which is ambiguous; that's one reason not to say it this way.
- All of us are staying away from the party.
which is unequivocally plural.
As for the motivational slogan
- None of us is as smart as all of us (note, no full stop)
It's intended to make you think, and it seems to have succeeded.
If you like, you can take it as short for
- None of us is as smart as all of us together are.
which is meant to contrast individual intelligence with groupthink,
and therefore is unequivocally singular.
tl;dr It'd be grammatical with are, but it'd also be a less efficient slogan.