Well, Silent Swordsman (and Silent Magician) only negate the activation of the card, without destroying it, so Geartown's effect would not trigger anyway, but, in a certain sense, you are right: if I were to use Ultimate Providence to negate the activation of Geartown, the card would be destroyed, but the effect would miss the timing.
Negated activations don't cause things to miss their timing, actually, so using Ultimate Providence would actually trigger Geartown.
Not sure there's a card in duel links that negates the effect of a Field Spell and destroys it without negating the activation either way, though. But if there theoretically was, it would miss its timing.
Oh, hey, Duel Links actually does have such a card. Destiny HERO - Dark Angel. If you activate Geartown while you control Destiny HERO - Dark Angel, Geartown'll miss its timing when Dark Angel destroys it, but won't if a card like Ultimate Providence or Magic Jammer is used to negate its activation instead.
>Negated activations don't cause things to miss their timing, actually, so using Ultimate Providence would actually trigger Geartown.
It would not trigger its effect. Look up Geartown's Official Rulings
I just tested in-game and I have to correct myself: if I negate the activation of Geartown at Chain Link 2 and destroy the card, the effect still triggers. Apparently we are not going by OCG Rulings.
...It works the same way in both regions. Negated activations don't count as a resolution for missing-the-timing purposes. (They still count as a Chain Link for cards like Lightning Punisher and Chain Strike, though, weirdly enough.)
Incidentally, the page you linked to proves this with its Magic Jammer ruling...