Thank you for the follow up, @RoxyRoxyRoxy! I gave that a shot - I emptied the cache via the Develop menu, fully exited out of all of my open browser profiles/windows, relaunched the browser, created a new empty profile to test this on. Success with that!
Came back to my default profile to sign in and it still doesn't do anything after clicking Log in. The only real difference between these two profiles is that the 1Password extension is installed as my password manager on my default profile and not in the new profile (as expected when creating it). I also did not have the 1Password extension set up in Safari, where signing in did end up working.
To that end, I installed the 1Password for Safari extension and was able to use that successfully to sign in. I suspect that because the Safari one is installed via the App Store and I'm using the Chrome extension, they will behave differently.
Please let me know if there is anything else I can try! 🙏