When I hunt, I carry a pumpgun with 2 low brass #8s and a high brass #6. This fired through an IC choke has served me well for several years. I've had a few living when they hit the ground, but not often. Shotgunning, as we all know, is more of a numbers game than a power game. Heave a pile of shot into the air and you improve your chances.
Pheasant don't really fly that fast compared to grouse. Maybe you are shooting too soon, before your pattern has a chance to 'open up'. Also, depending on the bird's direction of flight, you may be checking your swing leading to shooting a butt full of shot rather than up front where it does the most damage. I'd certainly look into choke tubes. I had my 870 bored out for choke tubes and my success went through the roof when I finally god ahold of a IC choke. The old modified was too tight for most of my shooting.