A string can be active and available in the GMC, but will not sync over to Gateway unless Account Approvers are tied to the Account String.
The Account String essentially "tells" the order where it needs to route in the system - so it will not sync over without having that information linked to it.
You can add approver(s) to the string under either the Assignments by User or Assignments by Role subtabs.
Please note: Account Approvers are the only roles that need to be tied to the Account String. All other roles, like Receiver, Requester and Department Buyer, are global. Once made for your department, they are set and do not need to be added each time a new account string is added.