In that example, "shouted" is definitely not invisible. I would probably have done it her way in that case.
That was a made-up example, here are some the editor used (i.e. added to my story):
"Good. You're here," said a dark-haired woman as Em walked into the Commissioner's office suite.
"I don't know about the where or the what, but the who would be me," said a voice behind him.
And an exception to my guess about only using it for minor characters, this example features the main character:
"Okay, so far the evidence supports our position," said Em.
As I said, I can't see either rhythm nor reason to her use of this phrasing, so I don't understand why she uses it.
I see the issue a little differently as to whether 'said' comes before the speaker or after. To me if the sentence leads with the speaker then 'said' is placed just before the words spoken. If it is after the spoken words then I think 'said' should come before the speaker.
Sorry, Rusty Ken, but I'm having trouble picturing what you're suggesting. Could you supply some examples?
Edit: Sorry, the middle example was my own creation. I used it to emphasis the "who, what and where", leaving who was speaking as a surprise for the following sentence.