Excellent questions Henrik. I worked with a number of accessibility audit companies in the past both in the EU and USA, who used both native (disabled) and non-native testers. Not once did they give us the feedback that the H1 had to be the same across the site. In fact, what they always praised us for was making each page's title different based on the page context, and then provide an appropriate H1 for that page. This is also in line with Deque and the IAAP, and whether Jason agrees with them or not, this is the accepted standard and this is what will be referenced in the court of law should a site ever get there because of non-compliance.