I used it on my own website not long ago actually..
I was having intermittent connectivity issues, and eventually couldn't connect at all. I checked the above tool which reported the site was online, but when I attempted to connect it would time out. Coupled to that I confirmed that I was able to connect using a proxy, something wasn't quite adding up.
I contacted my host and they advised that my home and work IP addresses had been blocked due to failed login attempts, which I still don't understand what caused this as there were no such issues from my end.
Long story short - it's a useful tool, even for your own websites!
I had my IP blocked by a host (I didn't know it at first) and couldn't get to the site and my friend told me to turn off wifi on my phone and see if I could get to the site that way! Success! But still bad because I don't do anything to warrant getting blocked.
It's extremely convenient for other sites. Personally, I have never had to use it for one of my own sites because usually, it tends to not have any problems in terms of hosting and uptime. But, when another site goes down, that tool is extremely useful to see whether it's a problem on your end or if the site is down for everyone else as well.