The Red hasn't had an Amyrlin since Bonwhin more than 1,000 years ago. Any woman who even remotely considered seeking a path to the Seat would look outside of the Red for a place for themselves that wouldn't nearly automatically block her ability to rise. Couple that with the fact that the majority of Amyrlins put members of their own (previous) Ajah into the key administrative roles such as MoN and KoC and you see how quickly paths to power for a Red are much more restricted than other Ajah.