They do that too, of course. But I have trouble believing that they always stop there. Look at their idiotic trojan PRNG, for instance (and if you believe their "What? That curve was chosen completely randomly! Of course we didn't construct it specially!" story, I have a bridge to sell you.)

You wonder whether it was some kind of crazy NSA intern project.

They prefer researching backdoors. The FBI implements them. Big difference. NSA researched and designed the clipper chips, then the FBI tried to get everyone to use the phones.

If by "implementing" you mean "sending an actual person over to plug in the recording device", yes. If by "implementing" you mean "contracting somebody to insert a cryptographic side channel in an operating system", then, very no.

which government agency is traditionally in charge of backdooring security systems?

FBI if it's like, a keylogger. NSA otherwise.