No, this isn't April 1st. I've found evidence on YouTube that there was at least one episode of an honest-to-goodness live interactive RPG show on a US cable TV network as early as 1982. Calling it RPG might be a touch strong - it's really "choose your own adventure", but with off-brand fantasy tabletop RPG trimmings. It's fascinating, more for its potential than anything else.

Prepare to scroll through 21 seconds of test card before you get to the content. With that warning in mind, I present to you 'QUBE Cincinnati - Swordquest', months before Atari would use the title for an unrelated, but coincidentally similarly doomed, series of adventure games for its 2600 console. The same YouTube channel has examples of other programming from a short-lived US cable TV network called QUBE which had simple live interactivity with its viewing audience through buttons on their remote controls. When the TC show prompts people to "touch now", it wasn't kidding. The cable network started in 1977, lost money hand over fist, and was shut down in the early '80s. Other interactive game shows on the channel made a virtue of featuring live polling data from as many as 168 families who were watching, despite having (in context) a big star at the time hosting.

The quality of the show is better than you might expect in some places, and towards the lower end of your expectations in other places. I suspect the gameplay was pretty heavily railroaded, though slightly more subtly than you might expect. I don't think its existence is at all well known, even among people who are interested in adventure games on TV.