Did you skip over post #35? It has commentary from the handbook stating why common trip would be needed.
That commentary is on 210.4(C), and Exception #1 of 210.4(C) applies to an individual branch circuit (just one load), so 210.4(C) does not require common trip in that scenario. The commentary on 240.15(B) excerpted in post #28 references 210.4(C) Exception 1 and agrees with respect to 210.4(C).
The remaining question is about 240.15(B) and whether it requires common trip, in particular whether the phrase "line-to-line connected loads for single-phase circuits" in 240.15(B)(2) means 2-wire loads only, as I believe you read it, or whether it covers 3-wire loads as well, as I am inclined to believe, and as the commentary excerpted in post #28 indicates.
It is worth noting that 240.15(B)(4), which covers 125/250V DC circuits, refers explicitly to "3-wire DC circuits" and so clearly would allow handle ties only for the (very rare) DC version of the circuit under discussion. So I have been pondering whether there is any reason for 240.15(B) to treat DC differently from AC, or whether the language used in 240.15(B)(4) is a clue to what is intended in 240.15(B)(2). I'm not seeing any reason for 240.15(B) to treat DC differently from AC.
So it seems to me 240.15(B) could be rewritten simply as "Circuit breakers supplying any load line-to-line with a voltage to ground in excess of 125V shall open all ungrounded conductors of the circuit both manually and automatically."
Cheers, Wayne