One thing I wouldn't do is force a confrontation with Auril on Grimskalle if the players are trying to avoid it. Apart from railroading, such a fight is likely to be inconclusive, since she can use her lair action to teleport anywhere she likes if things start going badly for her. Also note that her pet roc can drop people off the side of the building from a great height.
As for motivation, my players had already decided that they needed the Ythyrn Mythallar to end the winter, due to clues found in the Black Cabin and Lost Spire, so had common cause with Vellynne, despite not trusting her. It's perfectly reasonable for the DM, role playing as (intelligent and manipulative) Vellynne, to use whatever means of persuasion might prove effective on your particular PCs, and taking into account that she knows the PCs don't trust her, to get them to take her to Ythyrn. If the PCs refuse, she is likely to make off with the codicil and team up with Avarice, who will use the power of Ythryn to free Levistus.
As for why Auril will show up in Ythryn - the clues are on Grimskalle. She is a collector. She likes to collect and preserve things unchanging. She has "collected" Ythryn and sealed it away with her magic, which is why the PCs need the eponymous rime in the first place. If the PCs made a good impression in the trials she might be willing to tolerate them, if they take a "look but don't touch" approach, but Avarice hasn't been through the trials.