Hey Michael, sorry for the delay on this, we've had run-saving for the past month or so but it's been under pretty heavy testing. We've ironed out a few issues now though and if you're okay to use a still experimental feature we'd love to have you try it!
The "Quit to Desktop" button has been replaced by a "Save & Quit" button. When you resume you should be able to start (almost) right from where you left off. If you leave mid-battle you'll have to start that battle from the beginning, but in our testing so far it hasn't been too bad!
There's definitely still a few issues that come up occasionally that will do "interesting" things like cause a regular battle to bring you to the win screen.. Let us know if you run into anything weird and we'll use that information to investigate further!