I used to just get medium-fat chuck, but I much prefer Thomas Keller's suggestion of 50% sirloin, 25% chuck, and 25% brisket.
Also, I noticed the OP mentioned pressing it hard to get it thin. I think that is probably one of the most common things people screw up on with burgers, and certainly one of the easiest to remedy. Be gentle when forming your patties! It makes such a huge difference when you keep the "ground" structure/texture intact, and yet so many people manhandle their patties to the point that the meat might as well have just come from a food processor instead of a grinder...
Also, I noticed the OP mentioned pressing it hard to get it thin. I think that is probably one of the most common things people screw up on with burgers, and certainly one of the easiest to remedy. Be gentle when forming your patties! It makes such a huge difference when you keep the "ground" structure/texture intact, and yet so many people manhandle their patties to the point that the meat might as well have just come from a food processor instead of a grinder...