Great report. We were also in Tokyo with teenage boys, and we tried Yukemuri-no-sato. It's a huge and ultra-modern hi-tech public bath, not a tourist attraction at all, though they do get a fair number of resident foreigners, including teens. My two loved it - they thought it was weird, but sooooooo Japanese. And at $6 it's a bargain.
It's about 15 minutes from Shinjuku, near a station called Sengawa on the Keio Line. You need a towel - take your own, or buy one there. About 8 pm was a good time to go - you need at least an hour there. Food in the cafe is also very decent.