They are not saying don't use the words on that list in your book. Just don't use them in a tagline. If you use something other than 'said,' then ask yourself 'how can each word smile out of my mouth?' Or how can each word chide out of my mouth, volunteer out of my mouth... etc. Most of them cannot be done physically. As Moonshine told me, there are always exceptions, and there are places where nothing else you use can hammer that one important thought any better than a tagline.
Grammatical or not,l I think the site has a pedantic take on the whole issue. Anyone, with a modicum of critical thought, understands very well that "Hello," he smiled means that the speaker smiled as he said hello. That is different than saying, "Hello." He smiled. or He smiled. "Hello."
Not only does it create a clear picture of a man saying Hello with a smile on his face, but does so more efficiently than it would if he wrote out what a vast majority of readers would understand without a second thought: "Hello," he said with a smile. The quotation marks should make clear to any reader that he said, so including those words is redundant, whereas "smiled" draws a clear picture of how he said it and what his mood was in the greeting. A clear, concise, and efficient piece of writing.
Memphis Trace