My door responds as I'm passing my neighbor's house, and there's an empty lot between us. It's approximately 200' in a straight line from where I see the door begin to open. I have a Liftmaster operator. As a former overhead door technician, I will NEVER use a Genie. Regardless of brand, you should get much better range than a few feet in the driveway. I doubt the problem is the built-in transmitter, but it can't be ruled out. There are many factors, some uncontrollable, that affect receiver range. About all you can do is tweak the antenna, such as extending it to outside the garage. If it were me, I'd get a Liftmaster Model #312HM receiver and add it to the current operator and program my Homelink to that. You'd need a Liftmaster transmitter (remote) to program the Homelink if you have a GMT800. The GMT900 has it's own remote thing that's similar to, but not Homelink and doesn't clone an existing remote. The antenna on the 312HM uses a standard coax connector, so you could get a length of coax and move the antenna to outside of the garage. Liftmaster has an extension kit (Model #41A3504) with a purpose-made mount, but you could fab up your own with a double female coax connector if you wanna save the $20-$25. Personally, I'd buy the kit to save the hassle.
312HM:
41A3504: