The things that "start going really wrong" are listed immediately after that quote, and have nothing to do with the drop-off trajectory. Epic had problems with the ground stations they were using to communicate with their spacecraft. One of the two stations was offline because of a power failure. Then they "discover[ed] an unlikely incompatibility between their transmissions and the ground station hardware."

On a ride share mission, the primary payload determines the target orbit. If Intuitive Machines decides they want to go to a different orbit, then Epic has to deal with it. But they will be told this well before launch, with enough time to plan how many trajectory change maneuvers they need.

But even though they know beforehand how many burns they'll need, their exact parameters have to be calculated after launch once SpaceX uses the rocket's on-board guidance instruments to determine the actual insertion orbit. Usually this is within a few meters per second of the planned orbital velocity, but you need to know that last bit of error to figure out exactly how long the correction burns need to be. This doesn't change the overall maneuver plan, just the fine details.

The problem was that after launch, when SpaceX gave them the information they needed to calculate the burn parameters, they didn't have a working uplink to command their spacecraft to do the burn. So it just stayed on its original trajectory for longer than intended, getting more off-course the whole time.