Does a controller improve reaction times?
Your reaction times are what they are. However some games are designed to be played with a controller and the buttons
may be more accessible and easier to hit in a hurry than keyboard keys. Controllers also have a little joystick, D-pad, or rocker button which may be easier to use than arrow keys on a keyboard.
This brings me to another question: What the dickens is Quick Time?
Are you thinking of QTE (Quick Time Event}?
Essentially it's playing Simon as fast as possible.
While a cut scene is playing, the game throws a picture of a button push sequence on screen (blocking your view of the cut scene), and you have to respond as quickly as possible to mimic the button pushes that were indicated on screen. If you're too slow or hit the wrong keys or buttons, you fail the sequence and the cut scene shows something undesirable (or less desirable) than if you were fast enough.
https://en.wikipedia.org/wiki/Quick_time_eventA slomo/geriatric option in the settings menu seems more appropriate; however, that apparently is regarded as beneath notice; beyond the pale; de trop; and just not the done thing.
The problem there is that there wouldn't be much gameplay left without the QTE's.
Will a controller help me respond more quickly?
A controller isn't magic, however the button arrangement may be more accessible. Try one out in a store if you're able to. You should get one that fits your hands, not too large or small, with buttons in easy reach.
There are some games where a controller may help because the game was developed primarily for console and designed for use with a game controller (gamepad), and is clumsy with keyboard and mouse.