Where is the File Share Witness? If its in a 3rd location that remains accessible, then the DAG will stay online in the DR site.
If the FSW is in the main site and its down, and then the DAG loses quorum, you will need to do a datacenter switchover:
Either way, its on you to ensure DNS resolution and routing are online to handle the new routing to the DR site if the main site goes down.
Personally, I dont like DR sites. Both sites should be considered "prod" and your architecture should be designed to handle the databases and activity in both sites as "active". The FSW should be in a 3rd location to allow for automatic failover and HA.