It does notify by e-mail, sometimes to some people. I am a 'Follower' on every currently open job req, and a 'Hiring manager' on a few of the reqs. I'm not a 'posting owner' on any of the current reqs. There are 13 open applicants today, and I received e-mail notifications for zero of them.
This has been a recurring, long term problem with Lever -- and my company has used it for 3+ years (and still does, although we are finally transitioning off). It's to the point where it's someone's job to log into Lever every 2-3 days and nag hiring managers to respond to applicants -- and the hiring managers almost never know that they have applicants based solely on Lever.
Lever's preferred use case is also to have full access to Google Apps e-mail accounts, to provide end-to-end e-mail capture for when you talk to candidates. In theory maybe I want that behavior, but I really don't want to grant that control.
I'm glad that Lever works for you, but it really doesn't work for me. Using it is honestly one of the biggest mistakes we made as a company, and it's irrational that we stuck with it for so long (out of a sense that all ATSes suck so what's the point). It's never worked well for us, and it is fundamentally not the right product fit for us. I'm surprised to hear that it is the right fit for anyone, but it sounds like you have had a very different experience with it.
For my company, I can say with certainty that we have missed out on candidates, had an overall harder time in the hiring process, and had a MUCH harder time sourcing candidates due to its poor job board integration. That said, if you like Lever, don't let me stop you from continuing to like and use it.