Mag, its the terminology that's tripping those other mechanics up when discussing this subject! Every computer controlled engine has to have an "ambient air temperature sensor" on it because the engine computer has to have the air temp information to operate properly.
The MAF sensor abbreviation stands for "mass air flow" sensor which means it measures the "mass" of the air flowing into the engine so the engine computer can calculate how much gas to mix with the amount (mass) of air going into the engine. The air temperature sensor is part of the MAF sensor on the intake, it enables the computer to calculate the volume of air flowing into the engine. The computer calculates the exact mass of the air by measuring the flow through a throttle body of known size combined with the temperature of that air to give it the amount of air to be mixed with fuel for proper running of the engine.
The computer adjusts the mixture (air/fuel ratio) by varying how long it keeps the fuel injectors open, thus making the mixture "richer or leaner" which in turn has a great bearing on how the engine performs!
Starting a cold engine requires a richer mixture but if it's too rich the engine "floods" and won't start. The line between rich enough to start and "flooding" is fairly small. If the MAF malfunctions, the computer will not "know" how long to keep the injectors open to adjust the air/fuel ratio which can result in a no start/poor running condition.
Since you replaced your MAF and you still have an intermittent starting problem, you need to look at what else may be causing that. I would suggest cleaning the throttle body first and then replacing the fuel filter. If that doesn't work you need to check the fuel pump pressure. If the pressure is too low it can contribute to poor starts because the computer "thinks" it's getting more gas into the engine than it really is thus causing too lean a mixture for good starts. Too low fuel pressure first shows up in starting because it's more critical the mixture be correct when starting than just in general running after the engine is warm.
Fuel injection is a wonderful thing until it isn't