You are right that this join query should be equivalent. However, with InnoDB there will not be much benefit from your suggested index. In InnoDB primary indexes are clustered. Hence, any column may be retrieved from the B-tree of the primary index.
They could see a further performance gain by placing an index on routes (route_id, site_id) since the site_id could be retrieved from the b-tree and avoid a table hit entirely. But regardless, query syntax will not affect performance here.