The frightening thing is that your experience goes all the way through the system. I had two years of that getting to the point I could get an MRI scan with I estimate 50 phone calls pushing it through the system myself as my GP had gone AWOL. When I eventually got one it was in a privately contracted scan suite in a truck parked outside the hospital. Just one consultant radiographer and one nurse and super efficient. The nurse apologised for being 15 minutes late getting me from the waiting room. I said not to worry as I had been waiting two years. The consultant radiographer literally his jaw dropped open in disbelief. Just think my experience and your experience multiplied millions of times. Think of all the man years of literally useless phone calls being passed around.
I think you're getting close to the biggest problem - attitudes. I have some experience of "change" in industry, not a happy time but I remember two lessons: throwing money at a problem is usually money wasted, and the hard bit - you have to get the workforce on-side, pulling in the right direction and focussing on the end product.