Why is pinging an anycast address an issue? Most people, myself included, ping addresses like 1.1.1.1 to check for internet connectivity. It's a perfectly valid check, you don't need a fine grained test all the time. If it fails, then I go looking in detail. Most of the time it will pass.
Pinging a domain name could fail for a variety of other reasons, particularly if it's not a reliable site. Cloudflare's business is being reliable; few sites would be a better choice.
The problem is pinging 1.1.1.1 can end up hitting a really really close-by server. You could be having ISP issues and 1.1.1.1 could end up being closer to you than the issue is, so you get okay ping results but still unable to access resources on the other side of the issue