The Rule-Abiding Rebel nature of the Weyrwomen being "powerful" women whose only real power is political—the rest is just glorified kitchen duty, like keeping records and making sure the Weyr has enough food and medical supplies. It seems like Values Dissonance until The Master Harper of Pern. Weyrleaders are figureheads three-fourths of the time. Even in a normal Pass/Interval cycle, the Pass is fifty years and the Interval is two fricking hundred. Even dragonriders can only reach a hundred-and-fifty. And the Weyrleader is only in charge of wing-drills and leading everyone into Threadfall. In everyday life, outside of the awesome fighting? He can't do shit unless the Weyrwoman gives him permission, and when one of the largest dragons on Pern has her back, he'd better hope for a good relationship. Weyrleaders are only needed for fifty years, and might only stay in power for one year if the Weyrwoman hates him. On the other hand, the Weyrwoman stays in power until she dies, is crippled, chooses to retire, or her dragon stops producing clutches—which is generally thought to be way upward of forty years. Meanwhile, she controls everything outside of Threadfall—you're always going to need food, shelter, and medical supplies, but how often are you going to need someone to lead you into combat? Anne McCaffrey made an unwitting, but brilliant subversion.
Logistics are arguably the most important part of any military organization. Given that the Weyrs are at least quasi-military in nature, keeping track of supplies and records makes the Weyrwoman undisputably the most powerful person in the Weyr, whether Thread is falling or not. The Weyrleader's real power lies in negotiating with the atavistically paternalistic society of the Lord Holders.
Not to mention that Weyrwomen are normally fiercely independent—the ONLY time we know of that a Weyrwoman is dominated by a Weyerleader was R'gul with Lessa—even then, that was due to ignorance (R'gul was trying to control her, like I assume he did Jora, who was apparently a decent enough Weyrwoman while F'lon was around). So Anne actually set up the conditions well for a system like it to work, as well as showing the consequences when the conditions aren't met. Jora was controlled by R'gul, causing Benden Weyr to fall into a crisis, while Kylara, shirking her duties and being occupied mainly with intrigue and sex, caused absolute disaster for High Reaches.
T'ron and F'lar
There's a scene fairly early on in Dragonquest where T'ron and F'lar argue for a moment about an upcoming Hatching at another Weyr; F'lar is all set to offer some candidates he's found on Search, but T'ron shoots this down, stating that "weyrbred is best". At first, this just seems like T'ron being a tradition-bound old fart — which he is, but that's not the point — but with the publication of Sky Dragons, this scene now has a second meaning that actually makes T'ron's statement make sense! Sky Dragons once and for all retcons the infamous "Tent Peg Theory" and replaces it with the longstanding fan theory of "the dragon chooses, the rider complies". To elaborate on that, in a scenario where a straight male impresses a female dragon, when she rises to mate, if he ends up bedding another man as a result of their dragons' passion overwhelming them, neither he or anyone else in the Weyr will treat it as anything other than their duty to their dragons. Now, a weyrbred rider would have been brought up from youth to know this sort of thing wouldn't matter, but a holdbred rider wouldn't, and could end up freaking out at the prospect; hence why T'ron is against using them.
Alternately, T'ron could have been saying that because it's been more or less confirmed in-series that whatever it is that allows Impression is genetic (Hence the entire reason Fax wanted to drain Ruatha of all its bloodline). Weyrs would, by nature, have a far larger population of people who are capable of Impressing.
Menolly, grudgingly allowed to instruct her Hold's children in the all-important Teaching Ballads, the children must not think her original songs are part of the official canon, and (again grudgingly) her parents admit her work is so good it could be mistaken for that. She goes off score one night and segues into one of her own pieces — one measure's worth. Her Fantasy-Forbidding Father, listening nearby, takes off his belt and beats her senseless. He knew it was her own work after a few notes. That's how good his ear for music is. Can you imagine his childhood?