I think there's only one way to solve this — which I've been unsuccessfully advocating for at my current company — and that is paid voluntary on call schedules.

It creates an actual market for on call work where engineers can simply say no to the extra cash if they don't like work taking up their nights and weekends. If the company is having trouble with no engineers wanting to be on call the pay is simply too low and needs to be increased. It's a job like any other and should be compensated as such.

In the end I honestly believe it will be beneficial for the company not having engineers burn out so quickly. Compensation also clearly sets the expectations — if you're being paid to do it you'll take it more seriously.

Just my 2 cents