If that were the case, I think there wouldn't be that much discussion about the "website obesity crisis". E.g., see this post from another thread: https://news.ycombinator.com/item?id=15028741
Users who aren't using an up-to-date phone are probably not an audience websites are likely to make money from. If a website's performance isn't "good enough" on a modern phone, that will hurt the site.
Dark thought: maybe sites actually profit from a certain level of "bloat", if it drives away less lucrative visitors while not affecting the demographics that are most valuable to advertisers.