The other answers are right that in practice Cloudflare can't provide their full services as effectively without introducing this kind of security risk.
Roughly speaking, Cloudflare does two things:
They mirror your site, and can serve it from their own servers (their CDN). This way, if your site is getting hit with a DDoS, they can absorb the traffic and legitimate users will still be able to reach their services and see your site.
They do application-level traffic analysis to detect attacks and malicious traffic. This requires inspecting the cleartext of the HTTP traffic.
Obviously, the second requires Cloudflare to be able to see your traffic.
The first also effectively requires Cloudflare to be able to pretend to be your site, so you have to trust them.
Now in principle Cloudflare might be able to achieve some of the benefits of mirroring your site, without requiring as much trust in them. In particular, one could imagine a scheme where your site's top-level pages are still served from your server, but where subresources such as images, CSS style sheets, IFRAMED subpages, dynamic data, and other information is served from Cloudflare's mirrors. Perhaps your site's top-level pages could be served with a very long time-to-live so it can be cached for a long time.
In this way, a user who has visited your site in the not-too-distant past may still have the top-level page cached, and then the rest of the content will be served from Cloudflare's servers. This allows the user to still see a lock icon and your site's address in the browser address bar, and Cloudflare won't need to your SSL private key.
However, this kind of scheme has major limitations and huge disadvantages. If your site is hit with a DDoS attack, users who have never been to your site before won't be able to visit it. Worse, you'll have to change all your pages so that the URL of every resource is modified to point to Cloudflare's network instead of your servers. This is a huge burden and would make it hard for site operators to adopt Cloudflare's protection. A big part of the appeal of Cloudflare is that it is easy to set up: you just change a line in your DNS configuration file and you're good to go. The kind of approach I mentioned loses all of those advantages.
So, for all these reasons, the kind of alternative I outlined is probably not useful or attractive in practice.