I don't think it's a good solution, though. This is going to sound like an argument that you hear about privacy by people who don't understand why privacy matters, but I think on this issue, it holds a little more truth: To be honest, I think the people who are likely to care to do this are people who are advocates for privacy and are tech savvy, and people who have something to hide. I don't think it's terribly difficult, with the resources the global Intelligence Community has to build a profile and dump people into those two buckets with a fair degree of accuracy. Everyone who's not generating random garbage data would still be observable, so it wouldn't really change much outside of the group that's acting. But for those acting, suddenly there's a red flag that they can look for, and then when they see it, start building a profile to figure out if you're a tech person with an interest in privacy, or someone doing something they don't want seen, and act accordingly. In the mean time, we waste time generating garbage instead of just using good enough encryption, and making it so easy to use that people don't even have to know they're using it.
Now, if we could somehow implant babies with a chip that causes them to generate random noise (something babies are already pretty good at, mind you) from birth, that might be worth something. There's no profile building if the noise is just something that humans make by being alive.