Personally, if I was dealing with someone who wasn't already using strong unique passwords for everything, and didn't want the burden of having a password manager, I would absolutely recommend relying on Chrome's built in password manager and having to be signed in to Chrome everywhere. This might actually be one of the use-cases that triggered the 'auto sign into Chrome when you sign into a Google product in a tab' feature, which didn't go down very well in some circles.
I believe (re)using relatively weak passwords on multiple sites is a bigger risk to privacy than Google tracking every page I visit. I would assume they track every page I visit regardless of whether I'm signed in to Chrome or not.
EDIT: Note that I'm assuming this feature in Chrome is similar to the feature in Safari, where Safari suggests strong passwords, but this doesn't prevent those passwords from being stored in your password manager of choice.
If you start using Google's password manager with their auto-suggested long complex passwords, there is no way you can remember the passwords yourself. You'll need to be logged into Google Chrome to have access to the password. Therefore, you'll need to be logged into Google Chrome to log into any website. Currently, there is not way to export passwords from one browser to another, so if all your long complex random passwords are in Chrome, as a practical matter, you won't be able to use Firefox, Safari, or Internet Explorer, you'll be required to use Chrome.
The apprehension over the feature is not about security at all (in fact, this system is likely more secure). It's about control. If you allow Google to manage all of your passwords, then you'll need Google to do anything.
Coupled with clearing history, this should be fairly close to a much more stricter private mode.
