Communicating Value of PagerDuty

My organization just recently got a subscription for PagerDuty at the Pro plan level. This has been an uphill battle for me for almost 7 years now to get this as my current leadership believes all alerts and notifications should be via email and should always hit everyone all at once.
I came from an organization that used XMatters, so my experience with targeted alerting and triage is there, but it’s not far different in philosophy than PagerDuty. My argument is that we should only be alerting on critical items, and it should be using a workflow and start at the lowest level for triage then escalate as needed after triage. Instead of everyone getting an email all the time, proper incident management should be done where a ticket is also generated in our ITSM tool, then appropriate communications from the ticket go out at regular intervals to stakeholders. One of the arguments I often get is “We’re a small company”, which I disagree with, we solidly fall into the mid-size enterprise, but even then, size shouldn’t matter in doing things properly and responsibly.
Has others faced this kind of push back, email only and everyone always, and if so, how did you finally convince folks to embrace PagerDuty? 
My biggest push-back against email is with that, I can’t do any Inbox organization as Outlook mobile will only notify ion things in the Inbox, and I would have to create a notification that would wake me up at 2am so I can see if that email is just noise or real, and usually I get a few dozen emails overnight so that just becomes noise and a marriage killer. I can’t begin to count how many incidents were missed overnight and “dealt with in the morning” instead of properly triaged and addressed.