Wrapping the content in an iframe allowed the host to monetize our content by decorating it with ads. This would dillute the CTR on our ads, resulting in less money. Big sites enter into contracts with major brands for full page advertisements, which could include contractual requirements that theirs is the only ad on the page. Serving and decorating a site would create headaches for us to explain why their ads weren't alone.
The iframe also reduced engagement as reader would only read the embedded article and move on. An actual visitor of the site had a far higher likelihood to navigate to other pages. Higher engagement resulted in more money and some brand loyalty. This also dramatically hurt sharing as the browser URL wasn't ours, but the hosts (which I also hate about AMP).
Your site messed with a lot of the ways their business generated revenue, in an industry that was already struggling to adapt to "the internet".