While most view ASAP as a way to "get out of jail free" it is more effectively used as a tool to monitor unsafe trends. You do not have to screw something up (alt or nav deviation etc) in order to fill out an ASAP...even though that is probably the most common use.
I'm not sure if most ASAP programs work exactly the same, but ours has 3 people that review it. A company guy, a union guy, and a fed...if enough ASAPs get filled out for a certain condition (fatiguing scheds, sick call harassment, pressure to fly broken planes, etc) then more than likely your company's FSDO will put some pressure on the company to stop these practices. Also my ASAP program has an option where all you do is check a box and it will send out a NASA form as well. With the NASA form filled out then maybe the attention brought will be on a national level.
While it takes a while to fill out. It more than likely is worth it. I have no idea what the chances are, but ASAP reports may be used to help set up the new guidlines for crew rest and duty times...but at the very least it may cause your FSDO to pressure your company to stop some of the practices that lead to fatigue and pilot pushing.