Doesn't work out of the box, but can be fixed with a little tinkering on 10.14 Mojave
If you don't want a history lesson, the solution is at the end; this game is infamous on compatiblity issues, even on PC/Steam. For years people didn't know why you couldn't play the story missions (as it will crash during the first level,) but did work on the Mac version. It wasn't until the beginning of 2021 that it was figured out Adobe Flash (which is 'required' to play this game) was the culprit; as Adobe not only stopped supporting it, but outright blocked Flash usuage.
It was figured out the cutscenes were the issue; and even with Flash being disabled, the game would still load the cutscenes with a 'missing plug-in' texture. On PC it was figured out, but on Mac I had to do a little digging. Partial credit to drummer.clark on Steam.
1. Go to Finder and find Diner Dash Hometown Hero in your Applications section or in a folder for MacGameStore. Right click the icon and click 'show package contents.'
2. The folder you're looking for is hidden for security reasons. First click on the Contents folder then resources and enter on the keyboard at the same time Command+Shift+. (the period key.) You'll see a lone new folder called 'Game.'
3. You'll find the base game which has all the info for the game to run. Like last time, right click. 'show package contents,' and go into contents. Now go to resources, assets, downloads, _default, and restaurants.
4. Each folder are the locations in the game; all of them have one folder called 'Comics' with a .swf file. You can either outright delete the folders or just the files themselves.
5. The only issue you'll have now is during the loadup with the same 'missing plugin' - I'm sure there's another file you can delete to outright skip it, but you can just click anywhere to skip it (it's probably just the game company logo.)
When you finally get into the story mode, it runs perfectly! And all it took was a company taking away their product that used by hundreds of programs!