Actually it is very simple: adding Twitter to the mix adds one more "moving part". Shooting a simple email and then multiplexing it (via aliasing, for instance) to X recipients is easier and more reliable. Create an address like my-startup-site-is-down@gmail.com and forward to X other destinations from there (including Twitter if you want), and send your notifications to that address using standard 20+ year old protocols and tools, instead of some 3rd party proprietary stuff which was "released often, released early" and hasn't crashed "in months"