Anybody who rode a Schwinn ten-speed in the 70s or 80s remember those levers. There's an unavoidable safety aspect, which is that you don't have much leverage for controlling the bike when your hands are close to the stem under a hard braking situation. But I consider that to be folklore, as it's something I've heard about but have never seen actually cause a problem. Since they've more or less disappeared, it would be hard to get any contemporary info, but surely there are others who remember these levers or have them on older bikes.

The other issue that I've observed is that it's hard to get the brake levers into a position where you can actually get full braking force on the extension levers unless the components are actually high quality and everything is absolutely set up just so, from the levers down to the brakes and the truth of your wheels.

So the two open-ended questions are: Do they really work, and is it safe to use them? For casual cycling around town at low speeds, probably OK, but then at least in my case, the thing that worked even better was an upright handlebar and high quality braking system.