Who says CPs get their status by flying US?
I got mine last year mainly by flying UA and crediting the miles to my US account, since (1) I am fine with E+ for most trips and (2) had to take US for business trips, flying mostly out of CLT and wanted to be upgraded for free on those trips rather than being stuck in coach. If I were Silver, I'd fly UA a lot as there is probably a better chance of E+ on UA than an upgrade on US at peak times on peak routes. Even on business trips, I flew Ted a lot last year, with longer flights through ORD, instead of US direct flights whenever the only time wasted was my own.
If UA had significant operations in CLT, I'd switch accounts in a flash.