In "There's The Rub," Emily expressed frustration that she and Lorelai were not close the way Lorelai and Rory were close, but i didn't have the impression she was implying it was due to her failure as a parent. Lorelai might have seen that implication, but i didn't see it in Emily. I took it as one of her many moods, and this one desired being a buddy with her daughter. Besides, it was just another fleeting moment. If she wanted to see a failure in her parenting (or as a human being), she might have examined the way she absurdly accused Lorelai of forcing her onto the guy at the bar in the earlier scene. But she made no acknowledgement of that, nothing even implied.