The only thing they should be doing besides responding to incidents is work to stabilize the system to reduce the need to page people.
That's a fair way to make it work imo.
> If an organization thinks their systems should be available 24x7 they should staff people 24x7.
One way to achieve 24x7 coverage without anyone being structurally on-call outside of standard office hours could be "follow the sun" support -- have offices in multiple timezones around the world and run 3 x 8 hour shifts each day of support working their local office hours to provide 24/7 coverage. Still need enough staff at each site to handle rotation for covering weekends and for people to go on holiday and get sick or quit for new jobs etc.
I can understand why many companies would prefer not to pay for that if they can get away with cheaper alternatives, and why they might prefer to understaff and push the burden onto employees, but that's the company's problem, not the problem of the individual worker who trades their life by the hour in exchange for money.
Global ecommerce company letting people buy junk they don't need 24/7 at low low prices? An outage may frustrate customers and lose some sales but they might get most of the customers back in the long run, even if it takes them days to get the site to come back up and stay up.
Bits of the local electricity grid get damaged? Communications for ambulance dispatch go down and stay down? Those outages may directly cause fatalities or greatly increase the risk of fatalities until problems are safely isolated and service can be restored.
This lacks perspective.
Imagine that your system runs banking, healthcare, telecom, etc.
Of course someone needs to be on call 24/7. The best people to hold the pager are the engineers that work on the system.