How can I increase the responsiveness/speed of my stepper motor?
Your problem is that you want to go from a standstill to full speed instantly. That just isn't going to happen. As you have seen there is a limit to how fast you can go from stationary.

If you want to go faster you will have to accelerate up to that speed.

The simplest way to do that is to use a library that is designed for it, such as the popular AccelStepper.h library.