How can I make my iPhone 13 Mini design responsive in Figma so it scales properly across different mobile screen sizes without cutting content or leaving white spaces? I've tried constraints and everything, can someone please help me out?

There is no scale option in mobile present mode (it only available in preview mode), your frame should be exact size of chosen device.

But you can make your design responsive with constraints and auto layout, check Responsive setting in prototype mode and it will adapt to any device you want.