If you block third-party cookies, C has no longer has a reliable way to know that you are the same visitor on both requests. (Unless you're suggesting that C is stuffing a UID in the cache or something?)
C can already infer that. Google probably does that on their free CDN stuff.
you have unique combination of IP+UserAgent+extra Headers. That is enough. A and B does not even have to send anything. And this will continue to work even without cookies.
Requiring an IP address already eliminates cross-network tracking. For example, lots of people browse both on their PC on a cable/fiber connection and on their phone/tablet on 3G, with different IPs. They also often browse from their work network (yet another IP).
Same with User Agent: not useful if you're using Chrome on your laptop and Safari on your phone.
you have unique combination of IP+UserAgent+extra Headers. That is enough. A and B does not even have to send anything. And this will continue to work even without cookies.