Some cases where websites work very badly with a password manager (in my case the one built into Safari):
One case where the stupid website didn't accept the password that Safari suggested. Because it had some "special character" in it. What's annoying is that Safari remembers the password and suggests it again, BEFORE the site rejects it. Bummer.
Another case where the password is used in different places, and Safari cannot figure out that the two different places belong to the same site and should use the same password. What happens: On the second site, Safari suggests a different password which obviously doesn't work...
Apart from not working because of some stupid websites, it seems to be safe. The problem mentioned with password use on some public computer doesn't happen, because the passwords can only be used on my Macs, iPads and iPhones (but on all of them), but not on a random third-party device.