Not saying there should only be one H1 on the entire site. I'm saying that across the etnire site it is highly likely that the H1 should be the same on every page.

Thus why the site title / logo is the best candidate for it. If you're only going to use one H1 per page, and H2 means the start of a SUBSECTION of the h1, is making the biggest headline on the page going to make any sense structurally? Of course not.

It's a problem I see all the time and as someone who's vision comes and goes -- oft relying on a braille reader now to "help" -- multiple H1 and H1 that aren't the proper descriptor do make navigation harder.

The example I was taught that made it make sense in regards to H1 is how even though it's presented in larger text on the cover or front page of a book or newspaper, the title of the paper/book appears at the top of every page or fold-pair. That's the H1 on all those pages. The title of the book or paper. If you treat the website as the "book or paper" it becomes clear the name of the site and/or its logo is in fact that title that should appear on each page.