How feasible is it with medieval machinery?
Very feasible. Trebuchets were a thing in Europe around the 6th century AD and are very calibrable (yes, this is a word).
What, if anything, is wrong with this idea?
Murphy's Law, and not just how you imagined it when you read it.
Edward A. Murphy Jr. was working on project MX981 (a spiritual predecessor to Jackass) when he coined his famous law. The whole thing revolved around a very cranky rocket sled, meant to test the effects of acceleration and deceleration on stuff and PEOPLE.
His most famous volunteer, John Stapp (probably related to Ryan Dunn someway (also I just found out that he's my compatriot!)), seemingly enjoyed being used as a guinea pig on the rocket sled:
By June 8, 1951, a total of 74 human runs had been made on the decelerator, 19 with the subjects in the backward position, and 55 in the forward position. Stapp, one of the most frequent volunteers on the runs, sustained a fracture of his right wrist during the runs on two separate occasions, also broke ribs, lost fillings from his teeth and bleeding into his retinas that caused temporary vision loss; in one run he survived forces up to 38 g.
These things happen whether you are accelerating or decelerating very fast. But that was a rocket sled, which did it kinda gradually; Your human catapult will give its ammo all the impulse in a fraction of a second. All in all, you'd be squished inside if you attempted to fly like that.
Even if you have an army of supersoldiers who can survive that, another problem arises: in order to have close to no horizontal speed, you'd need to have the catapult close to the wall - where it becomes an easy target for pit, tar, and fire. Not much different from using the stairs on this regard.
But if you position your catapult far from the wall, the soldiers will land with A LOT of horizontal speed. It would be like leaping from a car going very fast on a highway.
Suppose you managed to solve the problems above: stairs are cheaper and have a better ratio of soldiers per second than catapults. Once you launch a soldier, it will take some time to ready a catapult for the next one. Even if you have lots of catapults, each soldier landing on the battlements would be easily surrounded and killed.
And that's considering you don't miss. Surely, medieval engineers could hit far targets with some accuracy, but changing wind conditions, inaccurate measurements and blind bad luck would ruin the day of someone who went too high or too low.
By the way, once you calibrate a trebuchet for a certain target, you need to always use ammo with the same profile and weight to keep hitting it. Different soldiers with different weights introduce a problem here. Maybe you can solve this by making the lighter ones carry stones as they are loaded in the catapult. But different people will have different aerodynamics.