I strongly prefer a standalone, offline-only password manager. It's the most secure. I personally wouldn't feel comfortable using one that can talk directly with a browser or where my passwords (encrypted or not) are being transmitted and stored elsewhere.

Password managers not only store website passwords (e.g. for shared drives or command-line tools), so they IMO shouldn’t be part of a browser.

Also, for those using multiple browsers, how would, say, Firefox access passwords or passkeys if Chrome stores them?

Having said that, requiring web browser extensions to enable integration of password managers IMO isn’t a good thing. There should be a common password manager protocol and a discovery mechanism that allows browsers to detect and communicate with the user’s password manager.

From a security perspective, one of the more important things is either a browser extension (or the password manager being built into the browser) to minimize the risk of the user putting a password into a lookalike domain/site. 3rd party password managers such as Bitwarden offer such browser extensions, while also simultaneously offering things like cloud sync support, something that you mentioned is a major quality of life feature for you.