Is there a way to generate random passwords in Safari, without iCloud? I know that iCloud Keychain has a feature where it offers to generate strong, random passwords directly in Safari. Some third-party password managers allow it, but I'd prefer to store the password in the OS X keychain, so I'm really just looking for a password generator, not full-blown manager. For other browsers there is addons that offer similar functionality. It's possible to do it in the regular non-iCloud Keychain Access app, but this is a bit cumbersome.

Is there any way of generating a random password directly in Safari, but without using iCloud keychain?
No there is no way in Safari directly,

You can only go directly to the Keychain  
(Spotlight -> Keychain // or via the Finder -> /Applications/Utilities/Keychain.app)  

and create a new Key (CMD+N).

You will get the window with the little key icon and when you click on it, it will show you a little popup with the Password Assistant.