The
entire point of security questions is that their answers are supposed to be things that are permanently stored in your memory, that you are physically incapable of forgetting because they are so ingrained. If you store these in a password manager, it is possible to lose them - and that is unacceptable.
These are supposed to be the very last line of defense for security, including if lose your password manager. As an exaggerated analogy, imagine that being unable to answer these questions meant your house, car, and life savings are taken from you. That is how important these answers are, except you're "only" losing one online account at a time.
Of course, it's terrible to use personal information that can be known to 3rd parties. It's also bad to reuse the same answers across multiple companies, as a compromise at one means you're at risk everywhere. The reason behind why security questions exist is a good one, but they don't offer enough security when used as intended (memorable, non-random data). The problem is there is currently no better alternative, short of requiring you to tie your legal identity to every account, and having to show up in person with photo ID to regain control of an account you've lost access to.
Anything relying on tech (like a password manager) is a bad idea for the general public. The average person does not have multiple off-site backups to guarantee that the information is physically impossible to lose.