Being more securely attached to your pedals will help your pedaling. Some platform pedals will accept
toe clips and straps to accomplish this.
Most riders eventually ride with clipless pedals and rigid shoes. With them, you can comfortably push harder on the pedals. I find the improvement especially noticeable when climbing hills. At the very least, try to ride with the stiffest shoes that will work with your pedals.
Getting down and aerodynamic is not easy for many. I don't do it for more than 10 minutes at a time because it hurts my back too. At the level you're at right now, it's more important to work on your fitness and confidence. The group rides sound like a great idea.
I agree with you about the groupset upgrade. It wouldn't make much of a difference in your speed, and there are many other things you can improve before making such an investment.