It seems if you’re using 1.1.1.1 as your resolver they (the website operator, not Cloudflare) are returning that IP as their A record. Sigh.
9.9.9.9 (and 149.112.112.112) are working great if you need an alternative public DNS server (https://quad9.net)
No, but thanks. I am of the opinion that if a provider is going to blacklist me because of my choice of resolver, then they just are not worth my time working around. I am quite happy with quad1 as my DNS provider, and these archive services don't provide enough incentive for me to switch.
If you’re trying to avoid the paywall, note that the NYT has removed the paywall on Coronavirus related articles. Including this one.