I have a buddy that is ambidextrous. He can throw both lefty and righty well over 400 and can shape great lines with both hands. He also has a usefull RHFH to about 300 and is working on his LHFH right now.
He used to ask the same question as you and his bag pretty much consisted of stable/overstable plastic. Until he started playing in the woods, that was working fine for him. Once he got into the woods, as others pointed out, you need to be able to throw a variety of shots that aren't always a stable hyzer. Example: you need to carry deep down a left-bending fairway where the best shot is a slow flip up RHBH hyzer with something less stable. Sure you can throw a RHFH or LHBH turnover, but if there are some fairway shapes that make that a less than desirable shot.
His bag is still mostly stable/overstable discs, but he does carry a Roadrunner, a Valkyrie, and a beat to flippy Roc for the occasional shots that require a slow turn.