Well one factor is the Seanchan begin collecting all the Channelers from the gene pool and tightly control them so finding the necessary genetic markers in two people would get more and more difficult as time progressed. Feasible to think that within a generation or two the ability would become very rare.

Seems to me that Aviendha's vision followed Parda because her line lasted the longest ... Perhaps Marinna and her decendents died during the war thus only giving Aviendha's vision one route to take. Alternatively it could have been completely random. But it seems like the Columns choose a path that follows major events and puts the individual into the most significant scenarios based on who was there. Nothing to say that Aviendha couldn't have seen parts of this future through Padra's family and other parts through Marinna's ... Only that we ended up in Padra's head.