Before iOS 13.0, the only way how to connect to BT classic device was External Accessory framework and the device had to be MFi licensed. Now that I can use Core Bluetooth to connect to BT classic device (custom manufactured device that should support GATT over BR/EDR), does manufacturer of the device still have to join the MFi Program?
No. GATT over BR/EDR is part of the Bluetooth spec and doesn't require Apple licensing. What iOS 13 added is the ability for Apple devices to talk to Bluetooth devices that support the existing spec. It didn't create a new protocol and isn't Apple specific.