For that, we need a mobile web app discovery platform that everyone knows about (and how many of these have been tried?) and can take people's money for you. I think if one of these had a few million for advertizing, there'd be a chance. (For that matter, how many people actually have looked at chrome app store more than once?)
For better or worse, the default app stores are convenient.
When the iPhone and iPod touch were released, that was exactly the solution Apple went with. They hosted a directory (www.apple.com/webapps) where everyone could submit their iPhoneOS optimized web app to be listed. Many of the web apps used local storage and you could save the shortcuts on your Springboard, just like native apps.
When the App Store came along, that directory became a wasteland. In June, Apple shut it down completely.
Apple’s iPhone User Guide was one of those web apps. Nowadays, it’s just a PDF.
Given that newer mobile OSes are going the route of web apps only, perhaps Apple was just 8 years too early with this concept (I wrote ‘8 years’, because I think we’re still 2 years off from mobile web apps being capable enough to rival basic native apps in speed.)