I'd probably argue that the sonatas (and other keyboard works) of Haydn, Mozart, and Beethoven really popularized the fortepiano. Mendelssohn had to help revive Bach from obscurity - so J.S. wasn't that popular. Of course, the mainstream composers knew J.S. Bach, but C.P.E. kind of eclipsed his own father for a while... partially due to be a bridge to the classical style away from Baroque. The WTC was very successful in pushing well temperament tuning, for sure - so I suppose that could be part of its enduring legacy....
It's really hard to say, at any rate. People were, to a degree, using some music on both... Beethoven suggested that the Pathetique sonata could be played on either.
I agree with you on the period instrument goal. Having played on harpsichords, clavichords, and modern pianos, I can't help but feel that piano removes some of the subtlety you get on a good harpsichord or especially on a clavichord. The touch is so different and even the fingering was quite different from modern keyboard playing. If J.S. wrote the WTC with a fortepiano in mind, he would have written quite a bit of it differently.
I suppose a piano version is certainly better than no version, though.