Edit: Is the answer simply "it doesn't, it gets flown remotely from Mission Control"?
They use a touchscreen so they don't need to have hundreds of buttons for things that aren't necessary to have around all the time.
Only questionable bit is how they do the docking without the screens.
It's interesting to wonder whether if Apollo 13 style fault happened, modern ships would be easier to reconfigure and rescue or harder?
If you've played KSP or read about Gemini 4, the relative motion can be unintuitive due to orbital mechanics. The SpaceX docking simulator puts you right next to the ISS with low relative velocity -- as it would be in real life if everything went as planned -- but it would be much more challenging if you needed to do it manually from a greater distance and relative velocity.
I think the main reason there are so few buttons on Dragon is that it evolved from an uncrewed vehicle that was designed to be flown from the ground, so everything is much more connected and software controlled than was even practical in the shuttle era.