Yep, sounds like the shifter's fine and the derailleur's been poorly set up.
How tight is the cable in the big ring? If you can pluck it like a guitar string, that suggests the high limit screw is in too far. Also, if it rubs in every gear, its alignment is obviously no good.
I'd suggest reading up and having a go yourself. It's a fiddly job, but it's not rocket science and perseverance should pay off.
It'd be hard to do a worse job than your mechanic, by the sound of it