Most cars in the US are automatics, and it does introduce a lag. I turn it off the moment I turn on the car.
It caused me to almost have an accident when I was in stop and go traffic, specifically because of the lag. The person behind me did not expect me to start so slowly right after stopping (I had to wait for the engine shutoff and then to turn back on).
I've had rental cars where it had an annoying lag and every start felt like a start from cold (ra-ta-ta-ta-voom), but in my current car there is no lag, by the time I've lifted the brake the engine is ready to go. An auto-stop start is also much quieter than starting from zero. I'm not sure what it's doing - this car has an ISG (integrated starter-generator), I'm not sure if that's the difference.
It caused me to almost have an accident when I was in stop and go traffic, specifically because of the lag. The person behind me did not expect me to start so slowly right after stopping (I had to wait for the engine shutoff and then to turn back on).