I can see it both ways:
Agile gives us a precedent for a similar but not identical ability, but was not designed by the PDT.
RAW says DEX instead of STR and calls out if an effect would deny STR then you can't use DEX either, possibly inferring just a numerical swap of the two modifiers, using all the CRB rules for how STR interacts with hands on a melee weapon.
-- or --
The assumed incomplete rule very well might be intentional. Add DEX to damage (period no penalty for off-hand) could be the deliberate new status quo for rogues. The much more common dual-wielding rogues get a bonus by eliminating a feat tax (Double Slice) and the rarer 2-handing finesse rogues pay for their more frequent crits and big dice with lower modifiers.
and isn't the whole point of unchained rogue more awesome with less hurdles to jump, but without the power creep of a whole new class?
FAQ requested either way.