Agreed on the value of an instructor here. It's better to have a coherent overall plan that impacts the plane rather than focusing on the plane as the sole primary variable you want to change.
That said, I've gone from steep to much flatter over the last two years working with a great pro, so I'll share. When I was steep, I really could only fade the ball, and I like being able to work it both ways now, which has become much easier with a flatter swing and more neutral path. Maybe even better than that, being more neutral allows me to adjust better on the course: if I'm hooking I can push towards a fade swing and straighten it out, or vice versa. This lets me avoid big misses on the course, which is huge.
Score-wise, I've gone from low to mid 90s down to consistently low 80s and am sometimes breaking 80 now, which was new for me just a couple months ago. I wouldn't say the flatter swing is the cause exactly, but I would say it's an important aspect of the larger changes.