The job of producer is to ensure things run according to schedule and budget. He/she should create or enforce the working conditions for that to be achieved. Of course it is not his/her fault if the director is not capable of achieving good results in a minimal amount of time
No, what you’re describing is the executive producer. The producer’s job is to oversee set design, story development, casting, shooting etc. and create a good motion picture. GL clearly didn’t allocate enough money and time for the film he wanted to make in the first place, given that he went back after the premiere.
Executive producer’s job is not ensuring the film to run on budget and schedule. Executive producer’s job is to get the founding for the film (getting investors, etc.). When the budget and schedule are set according to the allocated/available founds, it is the producers job to keep the production within those limits.
For a perfect being, you sure make a lot of mistakes.