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.
But does it give engineers a good incentive to improve product quality and reduce the number of production incidents?
Where I work we are not on-call. Nevertheless, I try to help the ops team when they encounter issues. This does make you improve logging and error handling since you know it takes a lot more time when it's difficult to filter logs for the interesting events.
Engineers not exposed to production issues and customers will never understand why you need these extra measures.
Our on-call system gives everyone an allowance per service plus additional time off if you're on-call on weekends and public holidays. You get both regardless of whether you get paged. Getting paid to do nothing is a great incentive for pushing quality, especially when you're on-call for more than one service in a given week.
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