I find your viewpoint rather confusing and I'm not sure I see it at all. The browser
primarily exists to traverse hyperlinked documents on the internet, which we call the "web". Though we've built plenty of "apps" now on top of hyperlinked documents, for the most part that remains arguably a secondary or tertiary reason for the browser to exist, and even if you could argue that secondary usage has rose to prominence to eclipse the primary reason for the existence of the browser, the use of the web for "apps" certainly has a lot less to do with Windows package management specifically and instead the convenience of simple, standardized client-server communications and development patterns that arise from that original primary usage goal of hyperlinked documents in a web of servers.
Web "apps" seem to only increase cross-platform interests, rather than serve to "smooth deficiencies" in a single platform.
> you can see why more competition among OSes would result in the browser being LESS important, not more!
We've seen exactly the opposite in practice: a ton of web "app" innovation occurred when there were a variety mobile "smartphone" OSes with reasonably modern browsers whereas development has shifted much further back to OS-specific apps as mobile devices have calcified into the iOS/Android duopoly.