> I think I'd have been happy with . Is this grammatical/natural?
It's good but " **** " is more natural.
> Is just a shorter way of saying this?
Yes. can simply take a noun, e.g. (order sushi / ask them to make sushi) and (leave them to do the rest). Usually we don't distinguish 'presidency' and 'president', so I assume can be translated into both of them and vice versa.