Thank you John for clarifying your concern and mentioning the browser.
First, make sure that Sync is turned on in Chrome. You can verify this by going to Chrome Settings. To do this, click on the three-dot menu and select 'Settings,' or you can copy and paste this into the address bar: chrome://settings/people Under 'You and Google,' ensure that Sync is on. If it's not, you'll see an option to 'Turn on sync.' If it's already on, the option will be 'Turn off,' which means you're all set.
If the issue persists, I recommend using Google Password Manager. Make sure it's allowed to save passwords and passkeys, and that the 'Sign in automatically' option is enabled. Additionally, enable Windows Hello to fill in your password. This way, once you're signed in to your Gmail account, you can use your PIN, fingerprint, or face verification instead of entering your password.
To do this, click on the three-dot menu and select 'Settings' > Autofill and passwords > Google Passwords Manager > Settings. Or use this direct link, copy and paste it in your address bar: chrome://password-manager/settings
I hope this information is helpful.
If issue persists please provide me a scenario so I can recreate it on my end.