I'm a firm believer in 'once you post it, you have set it free into the world and can't control how people interpret it', but that doesn't mean knowing an author's beliefs isn't useful for analysis. The author's beliefs and biases are going to find their way into the story, and the unintentional ones are, I think, even more important than the intentional ones--because if you aren't aware of a bigoted author's bigotry, and aren't bigoted in that way yourself, massive Weirdness in characterisation is just confusing.
My mom and I both found Harry Potter to have a lot of meaning and impact on us as a piece of art, and we also bond over it, as one of the few things we can bond over. My mom being a reader, and fond of mysteries, she picked up the first of JKR's adult mystery series, and read it, and there were things about it that... hit wrong, in a way she couldn't name. Then, the whole terf thing came to light, and my mom suddenly understood what it was that had seemed weird about the characterisations, about the choices that had made no sense to her. If she had never found out about the author's views, she wouldn't have been able to learn what those views translate into in art, and that's an important thing to learn.
Another example is Rudyard Kipling, Frances Hodgeson Burnett, and anti-Indian racism and Indian Orientalism in 19th century British literature. I grew up on the Jungle Books, Just So Stories, The Secret Garden, and A Little Princess--all of which are books by authors with varying degrees of colonialist and racist attitudes toward Indians and India. When I was younger, and didn't know about that, I inadvertently copied those views and put them in my own writing. I did not know I was doing that, because I did not know they were there, because I lacked any sort of first-hand experience, as I am not Indian myself. Once I grew up a bit more, and learned more of the nuances of 19th century attitudes and about those specific authors' other writings and opinions, I realised what had happened, and became able to separate out those tropes from those works, so I was able to avoid parroting them. If I'd never learned anything about those authors, I would have perpetuated more harm.
Sometimes, NOT knowing is a blessing--I read CS Lewis' Narnia series in its entirety as a very small child who had been entirely shielded from Christianity and so didn't pick up on those tones. This meant I could enjoy the books a lot more as pure fantasy. Now that I know, I can't read them again, because the allegory ruins the fantasy for me. So it's a double-edged sword, true, but at the same time--if I hadn't learned, I may have perpetuated those harmful ideas without realising that's what I was doing, and that's worse.
Or how about watching something: Recently, I watched the Dr Who episode "Boom", which was written by Stephen Moffat. Now, what I know about Stephen Moffat is that he is misogynist, and generally a mean-spirited person. If I hadn't known that, it would have been incomprehensible, some of the ways this story was bad; but knowing he's a sexist and overall rather mean person made clear why and where the story failed, and why for example the women characters acted so... well, to a non-bigoted point of view, so strange.
Sometimes you can't figure out what's off, sometimes you're too young, or sometimes you're so far away from that belief. It's not that you must know an author's personal life--the examples I quoted are all to do with an author's opinions which they have willingly stated in essays or other of their writings, or have to do with the time and place which they live or lived. This is not private stuff.
Because yeah, I am with you on "the private lives of other people are not our right or business to know"! I am I would say even more radical about that than a lot of folks. I aggressively do not gossip about the news on this court trial or that allegation or alleged crime. Because I learned from Horace Rumpole that Presumed Innocence is the golden thread which runs through a Just civilisation. We must presume innocence. We must stop oppression of the son-of-a-bitch because oppression is first aimed at sons-of-bitches, and it must be stopped at the beginning if it is to be stopped at all; but, an artist stating their opinions by writing essays, poems, or posts on public forums (such as twitter, facebook, tumblr, etc) is not their private life--in that case, they have chosen to tell us their opinions, and so it is not invasive to know them or talk about them.