Why would a PM want to know what some entitled silver spoon bag thinks of his policy anyways?
Probably because she has usually been quite popular with the general public in the UK. The thing is, if she HAD been willing to sometimes speak publicly about her views, she may have had more influence over the PM's because they would worry that if they did not go along with her view, she could say something negative about them through the press. I have to think that her commitment to never commenting probably diminshed her influence. But since all those conversations were private, no one can really say how much influence she had.
Contrast with Emporer Hirohito, another figurehead with no formal polical power, but who would sometimes express his views to the pubic through radio addresses. When the civilian and military branches of Japan's government could not agree that they should surrender after we dropped two A-bombs on them, and Hirohito privately told them they needed to surrender, the military had no choice because Hirohito would have gone on the radio and told the people that the military was consigning Japan to total annihilition. He was so popular that the military would have faced mass desertions.
The point being, that someone with no formal politcal power but with high popularity can have tremendous influence over those who do.