Some years ago, us sysadmins went from pagers to cell phones, with the alert system sending SMS via email, which is very close in function than what we used to get from pagers. I typically set the phone to some loud and obnoxious sound for a text, something that will wake me up. People who know me, know the very specific noise it makes.
One critical disadvantage we noticed right away is that cell phones don't last very long on a full charge. Even in the days before smartphones that last less than a day on a full charge, cell phones did not have the longevity of pagers, which could go a whole week or more on one battery.
Soon the practice was for the on-call person to carry about his or her person a cell phone and a spare battery. It was common to see an admin with the large bulge in their cargo pant thigh pocket for the cell phone and a slightly smaller bulge in other thigh pocket for the spare battery.
When cell phones with non-replaceable batteries started appearing, we got special dispensation to keep using a model with a replaceable battery. Now with external batteries being available, I don't know if this is still critical or just a convenience, but personally I prefer to swap out the battery rather than have a weight dangling off the phone.
If you decide to go with a pager, you should probably check on pager range, as it might have changed since the old days, as fewer people are using them. Whereas, I believe (but am not positive) that SMS works while roaming.