You better believe I would start streaming random data all over the internet just to be an asshole

Then, in the US, you have obstruction of justice and/or interference with police/peace/public officer.

I think it's more of a protest, or am I not allowed to email myself numbers?

You are, up until the point where it's cost a law enforcement officer time to determine that either the numbers are intended to waste their time or that the numbers are a hidden/unbackdoored encryption. Then you're GG SOL

You literally cannot prove it either way, you can't prove it's not enciphered data, you can't prove it's not random garbage, that's the point. You can say that 'we could develop a safe backdoored system that will allow only lawful decryption in the event of emergency' in the same way you can say 'we can launch probes made of candy to distant planets that will build cities and plant potatoes for us'. It's a fantasy. The keys will leak, criminals etc will still blend into the crowd.

How so? Couldn't we come up with a way of disguising encrpyted message streams so that they did not stand out? It would be more expensive, and given enough analysis they could probably detect them anyway, but it strikes me as an arms race.