HELL YA THEY DO
I am trying to do the Mega HyperDrive but it isn't working for me. I am running the Steam version and I am following the instructions you have posted. I click on a star that is very far away and then I activate the trainer and hold down like I want to warp there but all it does is return me back to my ship and not warp
This is the way it works here. Press M to open galactic map. Use the interface to get to another star. Make sure the info about that star pops up. Press the option key on for Mega Hyperdrive. Then simply click the mouse and that activates the warp. I guess if you have no warp cells left it might not activate. My ship doesn't even have a warp drive and it works.