They do save gas, but it really depends on how much time you spend stopped on a particular trip. The following article (with data from the SAE) says 7-26% fuel savings. SAE reports a 7% savings on the Federal Test Procedure, and a 26% savings on the New York City Cycle.
So probably barely any savings on a highway trip with only a very few stops that don’t last long. Potentially lots of savings on trips where you spend a long time waiting for traffic lights. It isn’t just some green washing to annoy people; it can save gas.
I’ve only experienced it in rental cars, and I just leave it enabled. Saving 10-20% on what I have to pay for gas seems a reasonable exchange for not having to do anything.
Sometimes the noise and vibration of starting is a little annoying, but I just get over myself, and don’t worry about it. The stupidest was a rental car in Texas, where at a stop light the engine would turn off, and then immediately turn on again so the AC could run.
If it caused a noticeable hesitation at a stop light, then I’d disable it, but for normal driving I find the engine is running as soon as I lift my foot off the brake.