Well, first off, if it's a modifier, it isn't an independent clause. Those are mutually exclusive. In the case of "experts say," the presence of a subject and a verb does not make it an independent clause. It's a tag that shows who said something, and it functions in the same way that a dialogue tag would work when we quote someone:
"Welcome," Dina said, "to the toughest week of your life."
"Welcome," Dina said, "to the toughest week of your life."
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