They control the reddest part of your page's heatmap. They inject JavaScript that does whatever they want. If you show up on a carousel result, for example, they inject JavaScript onto YOUR page that makes a left/right swipe navigate to a competitor. There's more, but yes...you've lost control.

"Caching your content," on a google.com URL, makes it their web site, not yours.

I disagree. Using cloud flare doesn’t make your site belong to cloudflare.

Cloudflare doesn't shove a header at the top of your page and JavaScript with unwanted behavior.

I mentioned the carousel problem in another comment. But the default behavior is suboptimal too. That injected header has an [x] button. Users expect it to dismiss the header. Instead, it navigates away from YOUR page, back to Google.

Does cloudflare change all your urls to theirs?