In my experience, you need more than a residence card to get a bank account here: you also need a certificate of employment from your employer.
I became a client of Sony Bank (yes, that Sony) but they would accepted "financial resident of Japan" which means people that have either 1) a work contract or 2) lived for 6 months in Japan.
It absolutely blows my mind why Japan Post is the only one to not have those restrictions.
>no appointment bullshit, no proof of residency asked)
Your residence card IS your proof of residency in Japan, hence the name. It even has your most up-to-date residence printed on the back.