I think this trans is related to the A4LD, so, if it is, the pump is mounted on the 'rear' side of the bell housing. There are two flats on the hub of the TC that engage similar flats on the inner of the pump gear. The problem you describe is as if the pump gear is not being engaged when everything is assembled.
I would measure the distance from the front face of the TC, where the bolts are welded that engage the flex plate, to the hub 'flats' on both TC's. If the new one were to have a different style threaded stud welded on, without a 'stop' washer, it could allow the converter to be pulled too far forward when bolted to the flex plate, such that the hub no longer engages the gear. Though the TC hub-to-face measurement might be correct, the in place position depends on the nose of the TC fitting the center of the crankshaft and the welded washers on the studs being at the correct depth.
In short check the depth measurement on both TC's for comparison. You definitely have enough fluid installed if it is coming out the vent, so you should not be worrying about 'pump priming' problems.
Do you 'feel' the hub engage the pump gear when you install the TC into the front of the transmission? We usually spun the TC a little as we fed it into the pump bushing & seal. [hub doused with clean ATF, bushing oiled and seal lubed a bit to protect for dry start.]
tom