Actually that is another sham lately by the Gaming Bullies that ruin our fine hobby. A framerate does not have to have anything to do with the controller response. It all depends on how it is programmed. Yes it would have if the controller only is checked after the start/end of each drawn frame for changes. Like how many times has button A been pushed in the time it took the system to draw this frame.
This however is not the best way to handle input, yes in the old days it might have when there really wasn't a way of having true multitasking running on your computer/console. But ever since multi core CPU's are a thing, or there are true multitasking solutions for single core solutions for single core systems. For instance when you buy a hard drive controller that isn't reliant on your CPU (rare these days, as so many SATA controllers are CPU dependant). But I'm digressing.
Now if you wish you could poll the controllers at 60hz, thus having the controller's changes read out at that framerate, if you then also update the state of the game on a separate thread. Running things like the controller, AI responses, physics calculation on a separate threads is actually far smarter to do than doing all of that after a draw cycle is complete. As such it doesn't matter if the game even runs at 15 FPS, as the input is handled separately and the graphics are only showing the state of the game rather than dictating the frequency of updates.
Now this isn't feasible for every game out there, some might actually wish for a tight coupling between the frame updates and controller input. But for most games that require speedy interaction, without having the speed of your GPU influence your game play input (like First person shooters), it is better to decouple them. So that during a small hiccup in frame rate your reflexes will still be read properly. It does however have a drawback, that if there somehow is a reason that your controler input is being read slower, like if the thread suffocates, it might appear that there is an even larger input lag
. But since controller updates usually are not that heavy on a CPU, that almost never is a problem.