No. An ounce of 6’s, 5’s, or 4’s from a 20 ga vs 1 1/4 ounces from a 12 wont have any practical benefit for your average shooter.
Stay away from pricey ammunition, it’s unnecessary and often the loads are pushed so fast that the pattern suffers. Pattern stringing is real but it’s overblown and a non factor.
Most people shoot 12ga better, not be because there’s more shot, but because the gun is heavier and swings smoother. Light 20 GA’s are wonderful to carry but stop as soon as they start. This translates into amplifying shooting flaws, the most common being lifting your head. A heavier 12 will likely have some momentum and swing through just a little longer vs. a light 20.
Don’t get me wrong, I’m a huge proponent of 20 ga afield. Just understand if you choose to shoot a 20, that your mechanics are much more important and it has little to do with the ammunition.