Clinic only allows pages to my phone

The hospital where I'm doing my residency has recently implemented a new paging system for clinics: through an app on my phone.

The old way it worked: If a patient had to talk to a physician from our clinic after hours, they would call the call center and be told that a physician would call them back shortly. In the meantime, we would get a page on our pager with the patient's contact info so we could call them back.

The new way it works: The pager is done away with. We now have to download an app on our phone that receives all our after hours clinic pages. We do not have the choice to use a pager.

I see multiple problems with this:

1. I like to leave my phone on "Do not disturb" when I'm sleeping so I'm not woken up by unimportant notifications. Now I can't use "Do not disturb" because I need to hear the ring tone when my phone gets a page.
2. I may not hear my phone's ringtone when I'm asleep.
3. What if I crack my phone's screen? My phone isn't as durable as a pager. I'll have to repair it, and while it's getting repaired, how will I get pages?
4. I don't want my employer telling me what I can and can't install on my phone. This isn't their phone to dictate that stuff.
5. What if I'm in an area where my phone doesn't get reception? I won't know that I got a page.
6. What if my phone runs out of battery during the day and I'm nowhere near a charger? At least with a pager, I can see when my battery is getting low and have a couple days to rustle up some batteries.

Would anyone else have a problem with this?