I don't think Amazon Prime or New York Times making it hard to cancel subscriptions is putting a dent in their brand image. Big companies can and will get away with it.
People should complain more and there should be more outrage. This is one of very effective way to keep the big companies in line. We are in a society that discourages complaining even if it is completely justified.
A screen that tries to offer me an introductory rate with a big subscribe now button and a small log in link. When I click log in,
A username form only - ie. no password form until I've submitted my username, which means I need to invoke my password manager twice. Then before I see the original article
A fullscreen modal wanting me to upgrade my subscription to get "All of the times" which includes News, Games, Cooking, Wirecutter and the Athletic (no pricing info), just a big Upgrade Now button or a small "continue without upgrading" link
before finally allowing me to access the article.
This is such an annoying pattern that was adopted by almost all websites. It almost always breaks password managers (requiring 2 invocations). I guess it "helps" people who can't differentiate between "log in" and "sign up"... at the cost of annoying everyone else.
Yes, it's true, that on the next screen it will tell you how much you saved using Prime and will offer to look at other plans or to continue cancelling. So this screen is sort of a dark pattern, I guess. Even then, privacy-oriented companies like Proton show a similar upsell screen on cancellation.
Overall Prime cancellation is very straightforward, compared to scummy companies that require you to email or call them to cancel membership. Unless the (unreasonable) expectation is that the CANCEL PRIME button should be front and center on every page on amazon.com. And clicking it should immediately cancel the membership without any confirmation.
Edit: the shaming element of the cancellation process was removed by Amazon long time ago.