Damn, I don't even know where to start. I haven't been to Aussie in a while, so perhaps things are different over there. Here, people put their shoes on the conveyor belt itself. I'm not sure how the germs migrate from the conveyor belt into the bin.
I would think that someone who is going to sit in a seat that has been sat in by a few thousand people and is not regularly sanitized would be less worried about what might in the bin that gets on his laptop. This is of course ignoring the fact that you've chosen to be stuck in a dry pressurized tin can that recirculates air that has been breathed by say 50-300+ passengers for up to the better part of a day at a time.
Do you honestly think they sanitize or even clean the handles of the overhead bins, the seats, arm rests, seat back pocket, doors to the lav, etc? I know modern airline cleaning crews are efficient, but you're in denial if you think they do anything but make the cabin presentable.
In terms of your worry about fecal matter, EVERYTHING in every bathroom you've ever been in is covered with fecal bacteria. There's even a mythbusters about it. In fact, you can probably find traces of it all over your house. It's everywhere. Run, hide, or just be an adult and deal with it. Do you think that the people around you all wash their hands before leaving the bathroom? They all touch many of the same surfaces you do. Moral of the story is there's germs all over your laptop, laptop case, ticket, passport, wallet, clothes, etc. Whether you use a bin or not really doesn't change this. If this is a real issue for you, I'd suggest looking up a 12 step program.
Of course, even in the press gone crazy sensationalize everything United States you don't hear of mass cries of people dying from infections caused by going through security. It's a non-issue.
Worried about stuff on your hands? Wash your hands. Worried about germs, don't touch your mucous membranes with your hands. Pretty easy. Everything is covered in germs. You can't change it. If you're a healthy person, chances are it won't be an issue for you.
Great suggestion though. Now we'll have to wait even longer and pay more to watch as people start lining their bins. Of course, most people will think that it's silly and elect not to do it. This of course begs the question of why you feel that the government has to provide this for you. Given that you have the phobia, why don't you provide the lining?