Yes, the church should apply for and receive its own distinct TIN. There is no reason why they should not have done so already.
My guess is that the issue of real estate property taxes on the property the church purchased may force some action. If the church operates under its secretary's TIN, it will be more difficult to claim exemption from property taxes. And if the church is required to pay property taxes, the secretary will have some explaining to do when the taxes paid are reported under his or her TIN.
Management is doing things right; leadership is doing the right things. Peter Drucker