After much hunting around to figure out why Chrome won't remember my Flash settings for HeroZero anymore, it turns out that this latest Chrome update deliberately doesn't save any whitelisted sites that run Flash. You either have to click to allow Flash on a site that needs it (like this game) every single time you open your browser, or you have to allow Flash everywhere as a default (really not recommended). Basically, Google Chrome's justification is that Flash is reaching end-of-life in 2020 and so they want users to be aware of which sites are still running it.
I'm putting this out here for anyone else who has recently updated Chrome or is about to update, so you know why it keeps asking you to allow Flash each time you start this game in a new browser session. And to save you time looking for a solution, because apparently there isn't one. At least it's only one extra click. Or just leave your browser running all the time so it doesn't forget the setting.