If you want to stay in this business you need to do the best that you can, working for/with people that have the same/better work ethic and motivations. Never stop losing sight of that or else you may become the same burnt out person that you are speaking of.
You do have to know your place in a kitchen, I have known too many "hot shots" that really were not, and ended up getting themselves in trouble for acting out of place.. If you know deep down that you are a valuable asset to the kitchen, and you are, then the chef should respect your input. You just have to know when it is time to move on or time to say "Hey Chef, you got a second"?
Do not listen to these people who think you should speak when spoken to, keep your head down because you’re "just a line cook". You are in charge of your own future and you are depending on the leadership of those you are working for to help bring you up the right way. If the person that you are working for is in the later part of their career and they are not close to where you want to be later in yours, maybe it's time to move on. It does not matter where you are working, there is always something to be learned, good or bad, so never stop paying attention to everything. I depend on my staff to participate in being the eyes and ears and I need their feed back on everything that has to do with making this environment an enjoyable/successful one.
I manage currently manage 2 operations and help out in both. I do 2 evaluations a year and I ask my employees to evaluate me as their supervisor. I value them as employees and I expect them to hold me accountable to make sure that none of us fall short of what the expectations should be.
If what you say is true about the chef...
1) Arrives to work at 12 noon walks in the kitchen and leaves to run errands until 3pm
Unless he is going to the Whole Foods Market to buy ingredients for the specials, I would say BS...
2) Has a nasty temper to the point servers are afraid to ask any kind of questions
Possible chemical imbalance or insecure about themselves as a person or chef.
3) Then claims to work 15-16 hour days (This would mean he is working until 3am)
If this is all true then I would not leave myself in the position to be given the advice of...
1) Suck it up, you are a line cook
2) You get paid to cook, not monitor what management does
3) If you worked for me and questioned my every move, you would be gone in a heart beat
4) It's not up to him to have an opinion on how the kitchen is run. He is a line cook.
5) Clock in, cook, clean up, go home
I would consider myself part of a team that prides myself on making the team better and more successful. If you truly believe that about yourself and you actions speak it as well, then I would not listen to 1 - 5 listed above. Successful people will surround themselves with successful people who are interested in chasing the overall vision of the business.
Keep your head up and do not settle for 1 - 5, brother...