Sure, Genius will then come along and copy-paste those lyrics, but that just means you have no reason to do it yourself.
Like - I want to buy a "Philips Electric Toothbrush". I could go and buy it on their own site. But I'll probably (sorry) go straight to Amazon.
Searching and checking out on Amazon is just so much mentally more efficient than navigating and buying from a site I'm not used to.
I agree bands should definitely upload their own lyrics, but comparatively, if I'm listening to a record and I want to know song lyrics or (sometimes) check what they're singing about - I'll just go straight to Genius because I know how it's likely to be there and it's easy to use (especially compared to lots of other lyrics sites). Band websites are not what they were in the 90s - if they have them at all.
After some Amazon anti-worker horribleness a year or two ago, I dropped my Prime subscription and now Amazon is my last resort for shopping. It turns out that as a web experience, it's pretty garbage these days. It's crammed with ads and dubious listings. So I think your "mentally more efficient" is what I'd call "habit". Ordering books from my local bookstore is just as quick and is definitely more pleasant.
I also don't think Amazon has spent that like you say on "getting that checkout process as frictionless as possible". What they've spent it on is magnifying Amazon's dominance and profitability. And a checkout experience is 25 years old at this point; it's not exactly a hotbed of innovation. My random local bookstore is using some perfectly solid package that does just fine. It's no harder to check out there than Amazon. And as a bonus, shopping is easier and they're always going to send me an authentic copy of the book.
Re "I'd expect that the overlap between Amazon users and online shoppers is very high." Honestly, I'd expect the same with Genius and 'people who look up lyrics often'. I mean - I can't name another lyrics website (apart from maybe songmeanings.net, which I'm not sure is still alive?) - and as someone who looks up lyrics a fair amount, Genius appears at the top of Google enough, and is far less spammy/shitty than any of the other random sites that I'll often just start there...
Allll this said, perhaps my analogy wasn't perfect. But I stand by that if I was in a band, and I wanted to share my lyrics - as well as my own site (assuming I could be bother to maintain than AND all the social stuff), I would make sure they were on Genius. Perhaps with decent explanations for things that I added myself.
genius.com is also associated with explaining the lyrics, which I suspect only a minority of bands with lyrics worth explaining would actually do.
Meanwhile, most of Genius's featured content involved artists explaining their own lyrics. It's a minority of artists, to be sure, because curated content isn't going to feature a majority of artists, but it comprised most of their branded work.
if Genius was actually creating more value than the licencing deals, then i'm sure the labels would have been happy to work something out. But what's the value for the content owners to have their lyrics on Genius?