Should screen readers be able to handle some unusual pages? Yes. Should websites design for accessibility? Yes.
Password managers aren't "Another thing to support" but "The only secure way to do passwords"
If your user can remember their password, they also likely: reused it elsewhere, have some pattern to it or minor changes that could be figured out from a email search in any password database, made it simple enough to be not secure.
I think this has to be highest benefit easy-ish thing you can do for someone to aid their computing lives in 2019 ...
Everything sites can do to help users undergo this transition would help them in the long run ...
The big password manager implementations need to do better as well — I don’t understand why iCloud Keychain doesn’t support generating random passwords that conform to the (horrific) password complexity checks you see out there in the world sometimes ... those sites are wrong to have such a broken feature but there are enough such broken sites out there that a clean workaround is needed on the password manager side ...
It would also be nice to have a solution for security questions built in — my solution is an OpenSSL command line for the random password generation and shared notes in which I record security questions and answers for sites. It’s better than actually providing real answers to security questions at least ... support for this functionality should really just be built into my password manager — the alternative likely thing is that a user will use actual answers to security questions for password reset all across the internet and this is not a thing the password managers should support their customers doing ...
Also, let's admit it, unless you do something really crappy like remove copy-paste, forms don't exactly "not work" with managers. Most of the time, you don't have to do anything special and it would work. Some just take a bit more time because you have to cop-paste it and not autofill. But people who are already using pw managers don't just stop using it (or start memorizing their passwords) because one site can't be autofilled. They just copy paste it, at worst, they manually input it while looking at the password from their pw manager of choice.
My line about "supporting" it is a bit off. I used the wrong words. It goes to say that you should support it. Again, you have to do something really out of your way to completely block off password managers from your forms so really, I think the norm is that they support it. My thought goes more along the line of whether I should be the one to adjust when the form works on some pw manager but not on another or when the pw manager can handle other sites properly and not mine. "Working" and "handling" here means it can be autofilled (most of the time).
Sites that want to display readable pages don't have to work on compatibility with Reader View; they can just provide readable pages. I use Reader View mostly to work around sites' intentionally user-unfriendly design patterns (articles unnecessarily split across multiple pages), and only occasionally to work around presumably unintentionally bad design (Kill Sticky does most of that work for me). To the extent that that's true, sites are likely to be interested in being less, not more, compatible with Reader View.
I use a password manager too and often wonder about this. Does this responsibility fall on the website's designer/developer or the password manager?
In one hand, I'd like my password manager to work on every site too but on the other, being a web developer/designer, I don't want another thing to support. We already have browsers and browser versions, and browsers and browser versions in specific platforms to keep track of. Do I want another layer of something to keep track of?
(This is totally unrelated but another thing I apply this question to is a page's/websites ability to support reading mode. You have straightforward pages that you can read wholly in something like Firefox's Reader View or Instapeper/Pocket. Then there are those pages that rely too much on some javascript library (sliders, read more, etc.) to display properly that gets broken when seen through reading mode.)