In family group photos do I need to list each person's name in order to have the photo searchable?
I believe - but do not know certainly - if you have tagged the photos they will be searchable by the names of those tagged - with the exception of living persons (protected space on FamilySearch for living persons).
Here are a couple of related Help Center documents:
Just to clarify about your living persons you have tagged in memories, you can make them private. If you are concerned about privacy for your living ancestors, the following article will be of help.
Hello @Richard M Smith
Some important things to keep in mind about memories in Family Tree:
Before you share any memories about living people, you should understand that all memories—even those with living people in them—are publicly viewable by default and potentially findable:
In the future, memories may be findable in even more ways. Please see our Submission Agreement before adding memories to FamilySearch.
Note: Individual memories can be made private on the individual memory page. Groups of memories can be made private in the Memories gallery.
If a tag is attached to a living person in Family Tree, the tag is shown only to users who can see that living person in Family Tree. If, however, the tag contains the name of a living person, but the tag is not attached to that living person in Family Tree, then the tag is visible to everyone who sees the memory.
Hoping this information helps answer your question.