Strict doesn't break links, it breaks sites that set "expected" cookies to strict. Those expected cookies won't be available in some situations that might have been expected, such as navigating to the site directly.
So if news.ycombinator.com cookies were set to strict, they will not be sent when I open the site, or indeed at all unless I was navigating from within the site (thus stopping cross origin attacks, but also stopping them being sent on first page load). If these cookies were used to identify a logged in user, the user will not be logged in as the cookie would not be sent. The link still works, but the behaviour is perhaps unexpected.
One solution is to have a trusted, strict cookie, and it is required for any actions that originate from the origin site - upvoting, new posts, comments, etc. You then have a second, untrusted, non-strict cookie that is used to identify a user. As long as this cookie is not used for any trusted operations, you have restricted the potential attack surface a lot.
None of this breaks links, it only breaks user experience expectations if utilised naively (like hacker news using strict cookies for their persistent login tokens).