The term producer can be taken in two different ways. Originally, the producer would be the middle man between labels, executives, recording studios, and bands/musicians. Today, beat makers are also called producers. Now, lets talk about who should be in the studio and handling the recording, mixing, and mastering process.
I'm an Audio Engineer, Producer and Beat Maker in Houston. Personally, I would prefer not to do all of those jobs on one project because the approach to perform each job is different; this can delay the completion of a project. As a producer I focus my attention the to things like the productivity, efficiency quality, budget, etc. The producer is the one that makes sure that the project is finished on time and within budget. An Audio Engineer's primary focus should be on the sound quality of the production from start to finish. Generally, engineers create space within a stereo field for each individually recorded track. So as an engineer I focus on uses technical skill to bring out the best artistic sound of the project. When I put on my beat making hat, the goal is to be a contributing member of the band; even if the only members are you and the artist. Kind of like a drummer or a keyboard player. Beat Makers are there throughout the writing process but shouldn't be in control of mixing and/or mastering unless that person is qualified.
It's seems to me that some people want to do too much when I come to making music which is why I try not to over work the process. Projects get done faster when the people involved focus on doing what they do best. People do their best work when they can focus on one thing at a time. At least, that's what I believe.