I think that the city flourishes when its grand institutions work together.
You need the it's in this sentence to show possession, regardless of your rewrite. It is a linking verb in this case, it refers back to the city as the subject.
Your re-write isn't wrong, but, the it's makes it clear that the main subject of this statement is the city.
The reason that we don't have -ing on work is because the subject that that verb is tied to is singular third person, it's 'the city' that's why work is conjugated to 'works', instead of being conjugated to ing for grand institutions.
Hope that helps, and sorry if I got too technical