First, many people log into non-Google services using their Google account. If you can't log into it from a third party browser, you'd have to keep using Chrome even if you didn't want to.
Also there is speculation that with browsers blocking third party cookies, Google might eventually perform ad tracking via the browser, via your Google account. Websites that live from ads like news websites will notice this and block users who are not logged into Google, as they don't add revenue. So eventually it might be possible that large parts of the web become inaccessible to you.
Also there is speculation that with browsers blocking third party cookies, Google might eventually perform ad tracking via the browser, via your Google account. Websites that live from ads like news websites will notice this and block users who are not logged into Google, as they don't add revenue. So eventually it might be possible that large parts of the web become inaccessible to you.