We just went through this at our volunteer station. None of this can be fixed without the Chief being on board, as it is his department and likely he and the rest of the crew have a long history together.
My department recently got a new Chief. He is top notch, as are all of the new officers. I will also add that they are all medics or EMT's and all but two of them are career fire fighters/medics as well.
We had the exact same scenario as you describe. The new Chief passed a rule that no one can sing onto a call until a licensed (FF1/2 or EMT-B) has signed onto the call. Period. Furthermore, all members who want to drive the rescue unit need to be at least FR's with an AVOC. Period.
This made a lot of folks angry, some quit... others just stopped coming to calls, but... a good number of the people (My self included....) looked at this as a training opportunity and enrolled in classes. I am currently starting my clinical ride time with a busy neighboring town who provides mutual aid to us.
We have dropped the number of calls we can handle, temporarily as an EMT needs to be available before we can take a call, but we have in one meeting increased the level of support our town gets even if it needs to come in the form of mutual aid until we get our numbers up.
So, bottom line... until your Chief is on board, I do not see how you would fix this without risking your department loosing their license. In my state your department is licensed to run at a certain level. We are a paramedic based Fire/EMS department. If someone had reported us to the state when we were in your situation the department would likely have lost our license to provide EMS services.
A tough road you are on.