Seconding what Lee in Richmond said, most of this is based on pre-plague work parties. Can still be the same with minimal contact and drop off from person to person. Typically we have specialists who participate in various parts of the project. Some of those like the hand sewing of binding and that is basically their part of the project. I'm a piecer, but I'm also a fabric collector/provider so I try to provide backs when called upon, or borders, or bindings or sometimes it is all three. There are always less people willing and able to quilt, so we usually take good care of our quilters and the top is already gone over once or twice before they get to them. It should have everything you need and ready for your attention.
It's ok for you to say "before we go further with this, this is what I am willing to do and this is what I expect from the projects given to me from here out". Include how many you are willing to do, the state they should be in, how big of backings you require, etc. And if you need/want help with thread costs, well yeah, you are already providing labor and equipment and it seems reasonable to me that could be donated to you as well.
So the incoming project would go to the intake person who would record who brought in what. They would examine it for suitability and workmanship and it would go into the "ready for quilting" or "needs something" piles -- if necessary there is a reject pile discretely tucked away somewhere. If there were reasons for rejection that needed to be communicated nicely and discretely, it would be handled by someone with that skill and not in a group setting. Needs pile broken down into basically "more workmanship" or "more fabric". Someone usually helped cover the fabric table and put blocks together with sashing or alternate blocks or maybe sketch an idea and kit out projects for others. I'm pretty good at that part and it's fun.
Doing it as a guild work party we would have various stations some more fun or desirable or more/less skill based than others. Although I don't like to iron, I believe in it and would often donate that time to the cause, always a busy station, keeping the tops as well as the backs ready for the quilters. Typically I did my own quilts start to finish on my own time, but this was my donated time so I go where the needs are.