Your way certainly makes more sense, and two carriages would cancel out the "vertical" effects.
But it's the horizontal effect that presents the problem. Yes, angular speed is constant at 15 degrees/hr. (360 degrees/24 hrs). But horizontal velocity has to increase from 1000 mph at Earth's surface to 6,880 mph at geostationary orbit (22,236 miles). The rising carriage would place a big "horizontal" force on the ribbon which cannot be canceled out by another carriage. As the other carriage is descending (and slowing horizontally), the ribbon will take on the shape of a huge "S".
Well, that depends what you're using it for. If you want to get into orbit, the geostationary point is your destination. The outer part of the stalk could be used, though, for slinging something out into deep space. Once you get past the geostationary point, it doesn't cost any energy to slide outwards. When you get to the desired point on the stalk, you just let go, and off you fly to Jupiter. Certainly the farther out you go, the smaller the weight limit for the payload, because of the stress on the stalk.
The rising carriage would place a big "horizontal" force on the ribbon which cannot be canceled out by another carriage.
True, but the horizontal force depends on how fast you move the carriage. And it could be compensated, in part, by having several carriages going up and several going down at all times. The angular momentum shed by a descending carriage would, on average, be picked up by an ascending carriage.
As the other carriage is descending (and slowing horizontally), the ribbon will take on the shape of a huge "S".
I've seen pictures where space elevators are depicted as having a shape like a car antenna in a stiff breeze, bowing in the spinward direction and then arching backwards. I don't know whether that's right, though. It would be interesting to see the correct calculation.
That's the height where you could just step out the door and hover right there because you would be naturally moving at the orbital speed for that altitude. But if you keep going up you would reach a point where if you let go of the handrail you would have escape velocity and would leave earth's gravity well.