Saw this tonight with my friend, specifically to see Julie White- and she does not disappoint. She is fantastic and much better than Weaver was (and I say this as someone who did not especially dislike Weaver).
White plays Masha less of a caricature and more of a deeply neurotic, vain, desperate person. She wrings laughs out of lines in a way Weaver didn't, but also brings a lot of nice shades to the character in the last part of the play. It's an entirely different performance and one that serves to strengthen the ensemble feeling of the production as a whole.
I really recommend seeing her in this, even if you've already seen it. If White had originated, she might've had a Tony nomination.
As for this production's success... it IS a success. It was a limited run, extended several times, and recouped. It won every major award for Best Play and the majority of its cast won nominations. It is an equivocal success and I doubt Weaver's presence had much sway on that.
"...everyone finally shut up, and the audience could enjoy the beginning of the Anatevka Pogram in peace."