You gotta remember that for a while there, when Dex was still playing like a Prez clone, before the Brothers came to the fore, and before James Moody & Sonny Rollins started playing Bird on tenor, Byas WAS bebop tenor. Once all that other stuff happened, his seemed to have become the road less travelled. Then he expatriated, ahead of the curve in that regard, really. And that was that. Every now and then the odd recording would pop up in America, but never a full album (not that I know of, anyway). He was a man forgotten, with a style seemingly no longer relevant.
But the influnce lingered on. You hear, say, Benny Golson, and you hear LOTS of Byas. You here any number of so-called "swing-to-bop" tenor players, you hear LOTS of Byas. Truthfully, you listen to some of Hawk's more "aggressive" things, and I think you're hearing a response to the technical/intellectual challenges laid down by Byas (and there's a 50s JATP in Europe thing w/Getz, Byas, & Hawk all together that seems to bear this out, at least to my ears). The public might have stopped listening, but a lot of tenor players didn't.
Sorry, but I don't buy the "second chair" bit. Don Byas was a MONSTER.