Not sure the minister should have posted personal info on the church fb page.
Ministers have a fair bit of positional power so, even without intending it, it may put some social pressure on people to buy wedding gifts. Sort of like the boss posting something on the work bulletin.
Putting it on a personal fb page is fine, because it sort of separates it out from the father's role as a minister. Can't really understand the issue about father vs mother posting info.
May depend on the church. I'm guessing based on OP raising the issue that church members aren't usually posting this sort of info on the fb site for their own family weddings.
As to the issue of no invitations or save the dates being sent out, I'm wondering if the OP is suggesting that releasing registry info to the congregation at large, many of whom may not be invited to the wedding, could come off as a bit gift grabby. Perhaps OP could clarify?