Manwe did declare previously that any being with any mortal blood would be mortal. This means Dior would've been mortal.
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Here Tolkien pointed out, that the descendants of Elros are mortal, following from his choice to be mortal. Would Tolkien have pointed that out, when the other choice would have led to the same result?
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No matter what choice Elrohir, Elladan, and Arwen made, their decision to be would permanent as well as settling the matter for their children. Elrond would be gone. A stipulation to the choice was that they leave with or around the time of Elrond, or forever remain in Middle-earth as mortals. So if E, E, or A chose immortality, their children would be born in Aman and have no choice - they'd be immortals. Likewise, if any of the three chose to stay in Middle-earth for good, their children would be mortal.
Source for that 'stipulation' is Appendix A in
Lord of the Rings:
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But to the children of Elrond a choice was also appointed: to pass with him from the circles of the world; or if they remained, to become mortal and die in Middle-earth.
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Macalaure, you have it right here: half-elves do not 'become' elf or man - they simply choose which fate they wish to follow. Tolkien always uses that terminology. In Letter 345:
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Arwen was not an elf, but one of the half-elven who abandoned her elvish rights.
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You also asked this...
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Elros chose to be mortal. Then he marries a mortal woman. Why didn't their children get to choose with a renewed mortal strain?
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This wasn't a "renewed" mortal strain...their mortal strain needed no renewing. His children needed no choice. The choice of immortality was given to Elladan, Elrohir, and Arwen in order to keep their family together if they so desired (as they would be mortal otherwise) - the same reason Earendil's choice was extended to Elros and Elrond.