Merrill was right about the mirror but incredibly wrong in how she went about it. Which is the point of her arc - her pride in thinking that she mattered.
Pride on her part, pride on Merethari's part, pride on the part of her clan, and pride on a rival Hawke who thinks he knows what's better for Merrill despite knowing nothing of the eluvian itself.
Pride is relevant in the entire arch and I won't deny this, but I see Merethari far more in the wrong than Merrill.
Once Merrill willingly left the clan, any and all responsibility to Merrill on the part of Merethari is now gone. Merethari's responsibility as Keeper would be to train up a new First to be her heir, see to the needs of the clan and ensure their safety near one of the largest templar strongholds in the White Chantry lands.
Considering said templars threaten the Dalish in Act 3 to convert, have one of their young tortured with fire, and by act 3, if you listen to the banter in he camp, some elves are even thinking of deserting the clan to seek another, I'd say Merethari is a failure in nearly all aspects of her duty.
Merrill did try to go through the proper channels. In Origins, we learn she knows some of the magic that Merethari uses on a Dalish Warden that held off the taint, and that magic was not amplified with lyrium or blood magic. Merrill took a single shard of the broken eluvian and asked Merethari to help her cleanse it and pool their magic. When Merethari refused Merrill learned blood magic to amplify the magic she already knew, and she outright admits that if she had access to lyrium she would've used that instead.
The devs have confirmed that Merrill did research on the eluvians and extrapolated all the knowledge she could from that shard and any lore she could get her hands on. Her eluvian is one she's building from scratch. Her eluvian is not the same as the one that was tainted by darkspawn in the Brecilian forest. And since she had the shard for a decade an never got herself or anyone tainted, I'd say she was successful in her endeavors.
She's proud, I won't deny that. There's a reason the Pride Demon in Feynriel's dream will target her above all other party members (Merrill, Fenris, then Varric, each betray's Hawke and then apologize for it later, whereas Aveline and Isabella will fall prey to the Desire Demon.) It is simply my opinion that Merrill is more in the right than Merethari, and I see Merethari as kind of a failure on all aspects of what being a dalish is supposed to mean, save teaching the city elves what was forgotten based on taking in Pol and how the City Elves treated her when she arrived.