Some good questions.
1 - One of the premises of a Password Manager is that they have One Job - they will be 10x more secure than a random site you entrust your password to. Indeed, if you lose your password manager password, they cannot recover your passwords due to the way the encryption is set up. Their site only stores strongly encrypted passwords which you password is required to reveal.
Secondly, two-factor authentication also ensures that any compromised passwords are useless alone.
Between these two aspects, the vault is secure.
2 - As mentioned above, being at a random computer is no issue - you can log into your vault from anywhere. Personally I cannot envisage a reason I would need to log into my secure accounts from a 'random computer' - which could be compromised with any malware, keyloggers, etc. But some people may have to and they are free to log into their vault to then log into their website of choice.