Edit: Additionally LastPass supports login to your LastPass account via password + OTP combination such as Google Authenticator and Yubikeys.

There are also peripheral issues with password databases, like the fact that they make the mere fact that you're using one transparent to anyone investigating your activities.

But ofcourse, that's matter of trust, even when they say that data is encrypted client side and they store only blob of gibberish. However I feel so relieved by using LastPass - not having to worry about remembering yet another password.

If you assume a key logger, you should also assume a mouse logger that captures a partial screenshot for every mouse click, as well as the possibility of capturing the contents of password fields (malware in the Windows 9x era would iterate through all OS widgets to find password fields and save their contents).

And one solution, whether it's backdoored or not, is still one target for bad actors to focus on (viruses, spoofing, etc).