which also means that this should be time that is paid on some hourly rate for the lower quality of live during the on-call time.
Also, there should be a rotation, with enough people to make this bearable.
There should be no rotation; there should be no on-call in the first place. If hiring staff to work normal hours is too expensive for the company, then the company has bigger problems than a service going down.
And having at least one daytime rotation agree that "well, you're expected to be on-call during business hours on Saturday" isn't nearly as crazy as "you can't relax at any time during the weekend".
There are plenty of people who would be happy with that and plenty more who would accept that arrangement if it came with extra compensation.
Instead, particularly at big companies, people get hired with some generic comp. package and then it's luck of the draw whether you have an on-call rotation or not, and if you do whether it's onerous or not.