LAX has four lines before you get to your flight area. There is the ticket check-in line, the luggage security check line, the I.D. line to go up the escalator, the security line where you get checked, and then you run to your flight area and hope they haven't changed the gate number while you are running towards your gate.
You will be begging people if you can cut in front of them and they all will say, "NO".
If that day happens to be an ORANGE ALERT day.... it will take longer to reach the airport due to a long line of vehicles. Plus, I have seen special lanes set up by security to check-out certain cars, so even if you are in a shuttle bus, there may be a traffice jam due to cars blocking the shuttle.
I would rather set a time for 2p.m. or later and have peace of mind.
I have had that experience and I live in Los Angeles.
The worse part is to see your plane still there being loaded with your luggage but they won't let you on the plane because you were 10minutes late and the gate is closed. Your luggage will be on the plane but you won't be. That happened to me.
Yes, Lady Luck may show up that day and everything will go as planned and you will leave on your flight at 12:30. Are you a gambler? Does stress give you more energy? Are you a high risk taker?