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.
The web UI is a bit old fashioned, and there's less functionality in some respects, but I find it easier to accomplish the basic tasks in. Really the only thing that I miss from Lever is the ability to go directly from one candidate to the next.
I actually disagree with most of what you're writing and I've been on it nearly full-time for the past few months. Lever has been awesome, coming from Greenhouse. It's also a much newer product, so as another commenter pointed out, it sounds like you missed some of the features that might have been more recently added.