It is important to note that all the shifts will do is mean that scheduling can only call you during your shift.
I could be on the 12 p.m. to 12 a.m. and still be assigned a trip that will keep me up all night.
I think we need shifts, but it will not solve all the problems with scheduling that cause fatigue.
The pilots are assigned a nine hour "do not disturb" window and any assigned duty period must include such a window in a 24 hour look back.
That means that they assign you a 5 am FRA, assuming a 8 hour block leg than from 1300 the previous day you must have had a nine hour "do not disturb" rest period.
Once you work through a couple of examples it is real easy.
When the FAA forced all the airlines to implement the look back for pilots, the airlines whined and complained like crazy, saying they would have to hire x amount of pilots. Turns out they needed to hire no one extra.
BTW, calling someone when you are called out on reserve may make you "feel good" for the moment, but will likely work against you as I would certainly turn off my phone, depending.
I would suggest a more powerful method is to write up every such incident in order to establish a paper trail and empower your leadership.