' All of these roll trippingly off my tongue, but 'today it's Thursday' does not. This sentence seems unnatural, but I can offer no reason except usage. My apologies.
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In my humble opinion, "Today is Thursday" is syntactically perfect, Today functioning as a subject / noun; and "Today it's Thursday is fine, too, with Thursday taking on the role of an adverb. This is confirmed by the fact that it passes the native speaker test (see above).