But it would not have mattered a bit whether Anne had or had not agreed to Mary taking precedence because the only one to decide in this matter was Henry. Henry never intended to legitimise Mary. Nor did he restore her to the succession until very much later. Even after Anne was dead Mary took precedence over Elizabeth only because she was the elder sister, but not because she had the superior rank.
But I'm one of those who do not believe in Anne's fall having been so sudden anyway. I actually do believe Chapuys's earlier reports about Henry having shown signs of being tired of his wife
Chapuys's reports are likely to have been true about Henry tiring of Anne, but at the time, as even Chapuys himself acknowledged, it was not necessarily significant. Particularly since Katherine of Aragon was still alive, and since Anne actually had produced one healthy child, there was no reason for Henry to contemplate getting rid of her early on. For if he got rid of Anne, there would be the expectation by many for him to take back Katherine, which he would not do. Of course, once Katherine had died and Anne had suffered several miscarriages, the situation changed. This supposed incapability of her to produce the heir was what really mattered in the end.
Concerning whether Anne's opinions mattered much - I think they very much did. In small matters: that Anne made increasing demands on Henry to obtain property of the queens of England that were in Katherine's possession, such as the royal jewels. Henry duly got them for Anne. In larger matters: Anne and her faction worked on destroying Wolsey for years, and eventually succeeded. One cannot say that Henry would have come to distrust Wolsey on his own.
But what is very important to remember, is that in the event of the King's death, Elizabeth would have succeeded him, with Anne becoming regent. Entrusting her with his kingdom was a huge matter - in fact, it says something about how Henry viewed her capabilities. Had she been entirely apolitical, then she would not have been entrusted with such responsibility. Of course, I entirely agree that Henry was the ultimate authority at all times,
but he was at times susceptible to the influences of others, in particular, that of Anne. While Anne's death did not alleviate Mary's situation, as she and her supporters hoped, that doesn't prove Anne didn't influence Henry in his treatment of Mary while she was alive. It just means that Henry's desire to break Mary into accepting his role as Supreme Head of the Church of England and denouncing the Pope, outlasted Anne.
Of course, in the end, Anne's influence obviously disappeared, but that's not to say that before her fall she had not wielded considerable influence with the King. Under her influence, he even read books that were previously banned. Here, Anne would mark particular passages for him to read with her fingernail. For example, the King even read William Tyndale's
Obedience of the Christian Man and declared it "a book for me and all kings to read."