Flash Player is a built-in component of Chrome (including Chrome on ChromeOS), and under normal circumstances, would stay synced with the browser. Chrome actually has two different updaters inside of it, and they've been working to use the Component Updater for Flash updates, because it solves a number of load and logistics problems. The transition has not been perfect, and they're still continuing to identify issues and resolve them.
It's a long-term win, but as we can attest to ourselves, writing installers for a large population means that anything that possibly can go wrong, will go wrong, given a population at Flash Player or Chrome scales.
Updates on Chrome are rolled out gradually across the population, which helps to soften the network impact of the entire Chrome user-base downloading an update simultaneously, but even then, I'd expect you to have received it at this point. We're on a monthly release cycle, so we're already halfway through the release.
If you go to chrome://components, you might be able to force the update. Just click on "Check for Update" and see if it updates. If it doesn't, I'd recommend that you reach out to the appropriate support channel for your Chromebook. Something isn't right there, but it doesn't have anything to do with the larger Flash and HTML ecosystem. Those changes are really about reigning in abusive advertising and tracking practices that marketers have deployed over the years (stuffing 10 invisible ads in a page, or writing a tracking beacon that sends a continuous stream of data eats batteries like crazy), and making it much, much harder to reliably deploy malware through insecure ad networks and watering holes. (We also continue to invest heavily in security hardening, in collaboration with Project Zero, Microsoft, and the larger security community, to apply the latest research and OS and compiler defenses as they become available.)
Hope that helps!
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