People will absolutely bruteforce random passwords. There are entire communities (like hashmob net, not sure if I am allowed to link it directly) devoted to cracking as many hashes of breaches as possible. Dictionary attacks will get you most of the easy passwords, but are quickly exhausted.
> Why do you focus on SHA-256?
I chose this hash because, thanks to Bitcoin, we know how fast specialized hardware to compute that hash can be.
> Is there some kind of statistics that this particular kind of hash is common among hacked websites?
It's not the most common. That would sadly be MD5. But SHA-256 is not rare either.
> They'll just hit my head with wrench until I unlock my iPhone, that would be cheapest attack on me, independent on password length.
I agree, rubber-hose cryptanalysis can be very cost-efficient. https://en.wikipedia.org/wiki/Rubber-hose_cryptanalysis Fortunately, many governments are opposed to this kind of cryptanalysis, but YMMV.
I agree that it depends on attack scenario. My scenario is: I expect website owners to find out about attack in a timely manner and disable all compromised accounts. Of course I won't reuse single password across different websites. Also I feel that most important websites nowadays require SMS or E-mail factor when logging in from another device, so this further decreases requirements for strong password.
And, of course, I don't expect to be targeted by government. They'll just hit my head with wrench until I unlock my iPhone, that would be cheapest attack on me, independent on password length.