Agreed. Logically, something like KeepassX (https://www.keepassx.org/) is the most logical, secure choice. I think a lot of people pick Lastpass and such for the convenience of browser integration, but I don't think that's necessarily impossible with keepassx - just so happens that nobody is really working on it (which is a shame).
This one in particular -- I use KeePass for my personal stuff, still; but at work, there seem to be a ton of logins we need to share.
Never mind sensitive stuff -- we get lots of use out of LastPass for managing the list of test and demo users on our site. We setup sandbox accounts (with various types of users) for potential customers. Each time the main logins to into LastPass, so if they run into problems, anyone on the dev team can help them out (with no other coordination required).
I've not been terribly impressed by LP's usability, honestly; but for quite a while they've seemed to be the only mature product in this space.
I've noticed Dashlane seems to be catching up here; I'm keeping an eye on them.
Beware, KeePass uses a weird custom key derivation function. LastPass uses PBKDF2 with a configurable number of iterations, a pretty widely accepted standard.
Maybe this has changed since I last checked but this and many other things seemed highly questionable on KeePass.
Back when I first signed up for LastPass, the killer feature for me was that it worked on my BlackBerry Curve. The fact that they made versions of LP for damn near every platform is what sold it for me.
I don't have a BlackBerry anymore, though. Now might be the time to jump ship.
I wouldn't consider Keepass the most secure choice. One of the most common attacks in practice is phishing, and browser integration discourages carelessly pasting your password into something that looks like your bank's site. The Chrome password manager and LastPass can help there, but Keepass does not.
