I do agree with you, but I have the feeling you have a narrow view of how these guys look like. Your argument was not too compelling. For all I know they could be giant gas clouds, silicon life forms, dark matter creatures, a fungi-like thing that spans the whole interior of a small planet and manages to shoot spores into space. Their motivations are not going to be in line with those of the average American. Their lives could span milliseconds, or aeons. Their brain could function at so extremely different rates than ours, that we or they wouldn't consider each other sentient. They could be highly intelligent* and/or propagate successfully without radio communication.

But I think we might be discussing semantics. When you say "intelligent", you probably have something specific in mind. Which probably boils down to "can do math". But a different intelligence could also grasp the laws of physics at an intuitive level, in the same way a dog can catch a frisbee. Perhaps they can "get" turbulence or higher dimensions,understand a trees as a whole, something we completely fail at.

So to me it's a stretch to think that we can find them by looking at error correcting codes. But then again, I have no better proposal.

Laser communication seems more likely for long distance communication. Omnidirectional radio broadcasts need a comparatively huge amount of energy.

https://en.wikipedia.org/wiki/OPALS

That's noisier, but it isn't a louder signal. We're talking about detecting alien species across vast distances. You can't do that by looking at a planet's light pollution AFAIK, because giving off this level of light is just not very notable — the sun reflecting off our planet is much brighter.