I did try out the new, still under development "Broken links" tab of the review page. And while it certainly points out a lot of truly broken links that need to be fixes, I noticed certain categories of false positives on Skeptics that might be connected to paywalls.
For several sites such as nytimes.com, nature.com, jstor.org, nejm.org, www.tandfonline.com all had multiple broken links reported, but all those links still worked for me.
For the NY Times I suspect that the reason for that is their paywall implementation, which allows to read a certain number of articles per month for free, and blocks you after that. I'd guess that the bot is hitting this paywall.
Nature is returning a permission denied status code for links to paywalled articles. While this is essentially a useless link for most users on many sites, such links are valid for many users on the scientific sites and many users there will have a subscription for this journal.
I'm not sure what the problem with jstor.org and nejm.org is, the preview/abstract is shown fine to me, and the site doesn't seem to return any permission denied or other error code.
I'm not sure how these cases should be handled for the link checker, the links are valid even though they are behind a full or limited paywall. It might make sense to exclude domains that have many false-positives reported from the tool completely, though that might also hide actual broken links in the future.