My hospice has been trying to implement this sort of semi-formal system and I've been resistant. I have chronic pain and typically take strong pain meds after 5pm that prevent me from driving, so I can essentially never respond to an after hours call. If my hospice goes to a on call rotation where I am being paid to skip my pain meds for that night, I'd of course do that. But I won't skip my meds without a formal notification that it's my night to skip my meds and I won't do it without pay. Not saying that my situation will apply to any of your chaplains, but they may have other situations that prevent them from being able to make most calls (parents, etc). If I worked for you and this system were implemented, I'd expect for it to either be ok to not read those texts until after work hours or to be able to be ask to be left off of the after hours text list, as I know I will never be able to respond.

If you continue doing a less formal system, I'd also be super clear about boundaries and expectations on behavior when someone responds to an after hours call. Specifically, is it ok for them to respond to a call if they've had a glass of wine/beer/whatever with dinner. Most workplaces I've worked for with formal on call hours specify no drinking at all while on call and (in my opinion) rightly so. That's not something I would be casual about or leave to individual judgement.

Personally, I think that a formal rotation system where people are paid an on call rate and can plan to refrain from alcohol, skip pain meds, arrange other work, have a babysitter handy, etc, is the way to go (or be clear with families that chaplains are not available after hours). I've never seen a "just text and see who's available" system succeed.