You are not getting it.  A seed phrase is not a password - its the master key to your whole crypto stash.  Equating it to a banking login misses how dangerous it is.  A password manager keeps all your different logins handy so you can easily use websites and stuff.  But I dont really see why you would save a seed phrase in there when you barely ever need it, if at all.

Your approach introduces a single point of failure that is entirely avoidable. Dont store your seed phrase online, period. Its not "an option worth considering"; its a major security flaw.