Like some other posters, you're thinking speed where responsiveness is more about latency.

Even an ancient pre-electronic planetary automatic can be built to shift faster than a human can row gears. Drag racing guys have been doing it forever, to the point that most performance car communities will have a "fastest <x>" and a "fastest manual <x>" if their vehicles were offered with both. Due to shift speed and a few other reasons it's just assumed that a properly set up automatic is going to be faster on the drag strip as long as the vehicle has enough power that the parasitic loss is negligible.

The responsiveness problem comes when you want a downshift and the transmission wasn't expecting it. It takes time to respond, even a DCT has delay in some cases, where a manual transmission is mechanically connected to the driver and thus has effectively zero lag other than whatever tolerances may exist in the various linkages.

That's a current Volkswagen T-Roc, odo 5000km, that drove me nuts this week. Then, I'm not saying I can shift quicker than that. Instead, I can shift 20s before stepping on the throttle.