I would like the idea, but when it comes down to the Ministry Officials, and the Healers, would they have to attend the Ministry Meetings? And then there is the Daily Prophet journalists, they have to go through the same application. Would they as well go into the ordinary Ministry Officials as well? I can see the pros and cons. I definitely see the cons of having the other employees, where someone could request a group change into that, and just say they are working and skip over the application completely. I can also see the pros because it would separate Ministry from Saint Mungo's.
With that being said, there could be a headquarters for Healers, to see who would take care of what appointment, see who would run Saint Mungo's as a whole, but that is just an idea out in the water.

I don't expect that to ever really happen, but it could be considered as well. Not sure if it would be used too much as well.

Reason why it might not be considered, seeing as other headquarters aren't really used, and there are no official rules to being a Healer.
If people need help to distinguish between a healer, whether or not they remain in the same group, they could just go into Saint Mungo's, in that pinned topic where it would be highly noticeable, fill out that easy form and a healer would post there if they want to take care of it.