Wouldn't work. These motors can only make like a billionth of an inch pound of torque anyway. As it is, with some motors and stock available gear ratios (12:1 - 8.5:1), you've gotta get a long distance to build speed. If top speed is your big thing, used "Kit Racer" gears instead of MicroSizer you could get like 4:1 - 3:1 and see if you can push that with your motors. Then, even that can be changed depending on if you have a 6 or 7 tooth pinion.
Your idea of doing that and trying with direct-drive would probably be lucky to even be able to move under it's own power, and if it would I'm sorry to say it would almost definately need a push-start.
You're talking 4 motors, 2 idler gears, and 2 drive axles. In order to power that many motors, gotta have more contact plates, wiring, and you'll probably have to go for more battery. Maybe a FET and more batteries so you can overboost your motors without burning PCB. All of those adds more weight, which needs more torque...
Plus you can't really synch any of the motors, so they're all fighting each other trying to turn slightly different speeds. Then there's more friction from more rotating parts. Blah blah blah...
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I dunno, if you're really bored go for it. If you prove me wrong I'd love to see it!
Sounds kinda cool!