Unfortunately for your system, if there's a network hiccup, you won't know why your jobs failed and the job will hang on the HTTP call. Also, you're paying for it instead of just sending e-mail alerts to yourself.
Critically importantly for this use case, if you are doing something critical for your business (say, backups or your daily billing run or sending reminders to thousands of people or what have you) and something goes wrong with your network and makes you unable to curl, you get told about that. If you instead do self-hosted email, and your network fails, you don't get told about that.
It's fail-open vs. fail-close. That's literally so core to the product it is alluded to in the name. A Dead Man's Switch doesn't go off if anything goes wrong, it goes off if everything doesn't go right.