The part of the elevator out beyond geostationary orbit acts as a counterweight. (62,000 miles sounds too far, however.) The pull that it generates is needed to counterbalance the tendency of the part of the elevator below geostationary orbit to fall back upon the Earth under its own weight. The key fact of the elevator is that its center of mass is at the geostationary point.
Start with a small object in free-fall in a geostationary orbit. Now let it grow a bit longer. Before it gets very long, the tidal force grabs it and it points towards the Earth. It's still in free-fall, but it's under tension from the tidal force, and its rotation has become phase-locked with its orbital period. The object as a whole is still in free-fall, though.
Now let it continue to grow until it one end of it reaches almost to the ground. It's under tension, but it doesn't fall to the ground and it doesn't fly off into space, because it's still in free-fall, as a whole.
This would create a tremendous force on the "overcooked pasta", depending on the mass at the end.
But that's mostly a tidal force. There is some additional tension caused by the rotational angular momentum of the object, which also has a period of one cycle per sidereal day. That's probably enough to forget pasta as a building material, but all the rest of the angular momentum has dropped out of the equation: we've already accounted for the orbital angular momentum when we stipulated that the object as a whole is in freefall.
Certainly, when that 13 ton elevator arrives at the "top", its mass is now adding to the stress on the pasta.
Yes, the stresses change in response to the position of the carriage. I imagine, though, that the mass of the "stalk" is huge compared to the mass of the carriage plus payload, so the position of the center of mass doesn't change very much. Moreover, you could operate two carriages in opposite directions to preserve the balance.
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".