Hmmmm.. I think that google thinks about sub-domain as a sub site, self-standing, totally separate, and can represent correlated topics.
Sub domains are treated as a completely separate site which is not affected at all by the domain itself
Wrong. Google could care less whether you use www or not. You just need to decide whether you prefer the www or non-www to be your canonical form and 301 redirect the non-canonical form to the canonical form of the URL you chose. If your most valuable content is on your home page then you have MUCH bigger issues than worrying about domain vs. sub-domain. Sites should have LOTS of valuable content on LOTS of pages. Most sites' home pages don't have that much "valuable content". The "valueable content" is typically on their inner pages. Sites with links from lots of sites to lots of different pages are seen as much more of an authority than sites who have most of their links pointing to their home page. To the OP: Google treats sub-domains as separate sites from the main domain. The sub-domains do not inherit any domain level ranking factors from the main domain like domain authority, domain trust, etc. Sub-domains have to start over from scratch building their own authority, trust, etc.