Shawn Michaels, Jeff Hardy, CM Punk, Christian, the Heavyweight bit was just a word added to separate it from the WWE title. It doesn't mean anything really. Just think of it as WWE Championship #2, WWE always have.
TBH I think of it as WWE Championship #1, but then again I started watching wrestling back when the WHC was on Raw, right before Evolution came about. HHH and the rest of Evolution did work with that title IMO. (for the record, I've went back and watched the entirety of the Attitude era)
Plus, I can't really bring myself to take the WWE championship seriously because it looks like it does, no matter how hard they push it and focus raw around it. It doesn't look like something ancient and prestigious for people to want to be fighting over.
And as far as the wrestlers that you've named, they're all at least twice the size of Rey.