Your observations about way more RHP than LHP is accurate. It seems that in general, RHP do much better. I expect that's because there are many more righty hitters than lefties. Switch hitters can be a good way to handle this, but pound for pound, they're usually a little more expensive.
Still, there are a lot of ways to win, and in any given league, you could find a surprising amount of lefties, throwing off your whole plan. Take some time to play more leagues, and keep gathering info like you are in this one. Try your switch-hitter strategy, and see how it goes!