So the total gear ratio of that gearchain comes out to something like 1119:1, though that's lower than you intended I suspect because you geared it down with a 9:57-or-so ratio on the end. The penultimate axle is geared at 7089:1 with respect to the first axle.
Now while that implies a rotation of one degree of the first axle is accompanied by a rotation of the penultimate axle through 20 or so revolutions, in practice that's not going to happen, because of that ton of friction you pointed out. The friction force acting to stop the last axle turning gets fed back as resistance to your pushing the first wheel, and it gets multiplied back up through the gearchain by the same factor - so the resistance to your turning the first wheel is 7000-times the resistance of turning the last wheel - plus 2000 times the resistance of turning the wheel before it, plus 918 times the resistance of the wheel before that... in total, assuming all friction forces are equal, you're fighting against a force 12000 times the force you need to turn just one gear - just to beat the friction. Okay, so you posit some frictionless maglev bearings in a vacuum, perhaps. But you still want to get these wheels into motion, so then the same multiplier gets applied to the force you need to accelerate the mass of each gear into (rotational) motion in the first place. Getting all those gears spinning requires 12000-times the acceleration to get one of them turning, so 12000 times the force.
And forget about putting a useful load on the end axle - let's put this thing in a car, say, and use it to drive the rear axle. We'll lift up the rear axle and get the engine up to 5000 RPM. Your engine is going to get that axle spinning 7,000 times for every engine RPM - 35 MILLION revs per minute! But when you drop your 1-ton car to get some traction, the engine's going to act like the car weighed 7000 tons...
You could also think of it in terms of energy. If that last gear spins at thousands of RPM, its got a huge amount of kinetic energy from somewhere. It must have come from your 'tiniest push'. Your tiny push has to transfer all that energy - so it can't be that tiny.
Gears 'distribute' kinetic energy in that they allow you to take a rotational motion on one axis and generate a rotational motion on a different axis. That can be a parallel, offset axis, as in this case, but it can also be at a different angle, using angled or crown gear teeth. So yes, that's one use, certainly. They let you reverse a rotational direction, too - turning clockwise into anti-clockwise.
But the main thing they do is let you trade angular distance of motion against angular force, or torque - same as a lever does, and much like how a pulley system lets you trade off linear distance of motion against linear force. I can make a gear system that multiplies the effective force I can exert by 50, at the expense of my having to rotate a crank on the input shaft fifty times for every time I want the output shaft to rotate. Or I can use it the opposite way around and make an output shaft spin fifty times for every rotation of my input shaft, at the expense that I have to exert fifty-times the force to overcome any load on the output shaft.
Now while that implies a rotation of one degree of the first axle is accompanied by a rotation of the penultimate axle through 20 or so revolutions, in practice that's not going to happen, because of that ton of friction you pointed out. The friction force acting to stop the last axle turning gets fed back as resistance to your pushing the first wheel, and it gets multiplied back up through the gearchain by the same factor - so the resistance to your turning the first wheel is 7000-times the resistance of turning the last wheel - plus 2000 times the resistance of turning the wheel before it, plus 918 times the resistance of the wheel before that... in total, assuming all friction forces are equal, you're fighting against a force 12000 times the force you need to turn just one gear - just to beat the friction. Okay, so you posit some frictionless maglev bearings in a vacuum, perhaps. But you still want to get these wheels into motion, so then the same multiplier gets applied to the force you need to accelerate the mass of each gear into (rotational) motion in the first place. Getting all those gears spinning requires 12000-times the acceleration to get one of them turning, so 12000 times the force.
And forget about putting a useful load on the end axle - let's put this thing in a car, say, and use it to drive the rear axle. We'll lift up the rear axle and get the engine up to 5000 RPM. Your engine is going to get that axle spinning 7,000 times for every engine RPM - 35 MILLION revs per minute! But when you drop your 1-ton car to get some traction, the engine's going to act like the car weighed 7000 tons...
You could also think of it in terms of energy. If that last gear spins at thousands of RPM, its got a huge amount of kinetic energy from somewhere. It must have come from your 'tiniest push'. Your tiny push has to transfer all that energy - so it can't be that tiny.
tl;dr: no.