We know nothing about Seanchan pre-Hawkwing. They may have had the Aes Sedai in the gene pool.

Also suldam are part of the pool, unlike Aes Sedai and Kin so they haven't shut them out.

So you may have had 2000 - 2800 years (the range covering up to the end of the consolidation) of full female involvement from the Breaking to Hawkwing that didn't occur in the westlands. And since then still non-Sparkers potentially breeding.

It may even have been that until Hawkwing's day that given the Aes Sedai there were busy contending amongst themselves, that they left alone male channelers until they became an explicit threat.