Browsers can of course be non-conformant on other interesting issues. Imagine if a browser came out that allowed the Unity Web Player to work. Instantly, people wouldn’t need to use the App Store to purchase games. Instead, they could just use the browser.
For the 2nd, it sounds like there needs to be a way for the web platform to evolve for things that require engine modification. I wonder if the web will break down into smaller standard components and new browsers can be built from those.
With regard to your second paragraph, I believe you have made a category error between "the App Store" and "the browser." The browser is not a store.
I can think of a few possible reasons:
1. Chrome (as in the browser) came out after the first iPhone was released (2007), and around the same time of Android's first release (2008). Chrome OS didn't come out until 2011. Android's app ecosystem started developing before Google had all that much influence on web standards, and likely before they realized that it would be a good idea to have that influence, or even knew that Chrome (the browser) would become as popular as it ended up being. I'll also note that Chrome OS wasn't anywhere near as successful as Google hoped it would be; while that doesn't invalidate Chrome OS's app model, it doesn't validate it either.
2. Android has done a decent amount of copying of iOS features (I say this without judgment or malice; I've been a generally happy Android user continuously since 2010 and have no axe to grind). Not having native app development on Android could have been seen as a possible market disadvantage when compared against Apple.
3. Google may have actually desired more Apple-like control over what runs on Android, or at least wanted to allow for the possibility that they'd want it in the future. That's harder when you base your app platform on web standards.
Really it's just code for "let me, a web developer, target Chrome only, so I can just tell Safari users to install Chrome."