I'd say, the blurring of work/off-work hours is one of the greatest coups of industry. Having someone on-call means they don't have to hire someone else to be there during "off-hours". I understand it adds costs --yes, pass them on [but someone else which cheap out and just have on-call and undersell us: pass laws to eliminate the choice.
I'm no fan of the industrial age with its "dehumanization"/abstraction of labor, but I do appreciate the hard delineation between work hours and non-work hours.
I had a leash (pager) in 1992 that was tugged often. The fur really flew if I didn't call back immediately. eg I almost got fired because I was in the shower at the gym (at 7:00am) and took 15 min to call back.
From what I can tell, since pretty much everyone I know has done a tour of duty at Amazon or equiv, things have only gotten worse.
"means they don't have to hire someone else"
Natch. But I see it more like someone else's failure to plan creates an (recurring) emergency for me.
The only difference for a website is that no one but you and your customers care if it goes down, so it's up to you to provide the manpower (unless you outsource it to IaaS/PaaS providers).
Note that air control towers are staffed with air traffic controllers, not aeronautical engineers. Also note that they're being paid to be there, and they don't ATC from their laptops in bed.
Likewise, operations is different than development.
(Granted, you can have those roles fulfilled by the same people, but that's your choice to make air traffic aeronautical engineers. YMMV)
Go read "Site Reliability Engineering" by Beyer et al. That's the clearest description for how on-call should work, and the way to get from Point A to Point B without burning anyone out. On-call is almost always necessaey, but should be thought of as high-deductible insurance. If you need it, use it, but just realize it's cheaper to prevent the problem than to fix it when things go wrong. And if you keep using it unnecessarily, your carrier will drop you.
Boeing doesn't require an aeronautical engineer in the cockpit of every 787, because they engineer their product to be operable. Have smart people build your infrastructure so that junior people can operate it, or else you wind up with unstable, burned-out, formerly-smart-people.