You are describing signal jamming tech (single-channel, sine-wave) that is decades old.
Modern EW platforms are capable of covering entire RF bands, adapting and following hopping schemes, and efficiently spreading their energy over seemingly pseudo-random code-schemes.
In the end, there's only so much you can do with modulation techniques - it comes down to signal strength - and the inverse-square-law pretty much says that who-ever gets closer wins.
The control signal from the US base comes likely via LEO sat-link or over-the-horizon AWACS-type platform - both of which are going to be hundreds of kilometers away. You're not going to need "absurdly more powerful" anything to interfere with that. I have a wide-band I/Q generator able to modulate any mathematically describable code-sheme - which I could then hook up to our MIL-STD-461 susceptibility testing-chamber-amp - and knowing something about the signal band I could easily get the right high-gain antenna to track the bastard off the sky... and all this is with off-the-shelf COTS equipment!