The scanning frequency is not that significant. The response time is more limited by something called "debouncing".
When a key switch is pressed, it does not actually change state from open to closed in a perfect way. Instead, it often "bounces" between open and closed states for a little while until it settles. Another issue is that short spurious positives may be caused by static electricity. Keyboard microcontrollers will therefore have to delay reporting each key press until it is sure that the key reports a steady state.
Debouncing is almost always implemented as a counter or a buffer with a delay - and that delay is fixed. The delay is chosen based on the characteristics of the particular keyboard switch.
By increasing the scanning rate, the microcontroller will only gather more samples for debouncing each key. As you increase the scanning rate, your response time will only approach the debouncing delay; It will never surpass it.