The "under sink" pumps you've referenced do not need a vent as they pump the water directly out of the sink and act as the sewer gas seal for the sink's drain. That would be the best choice in my opinion. But as CT18 said, the video shows the discharge being connected to the overhead sewer in the horizontal. That could cause ongoing problems depending on what is upstream of the connection point and is definitely not best practice. The connection to the sewer line should be vertically from above, or as a minimum 45 degrees off vertical from above.
Now if you choose to use the Zoeller system, a small sump with an internal pump, the sump does need a vent and an AAV is not acceptable. This vent provides air to the sump as the water is pumped out. Without it, the air would have to come through the P-trap and the pumping action would siphon the P-trap to a low level. Being an infrequent use sink, any residual water could evaporate, and you would have an opening to the sump. Granted, the sewer gases would still be sealed from coming out of the sink exactly like the "under sink" pumping system via the Zoeller check valve, but that is not per the design of the Zoeller system and is not an acceptable installation.
Now as you have another sump vent 50 feet away, and if the vent on that sump is properly routed outside, you can connect the new Zoeller sump vent to that vent. The vent line should still be sloped 1/4" per foot towards a drain. Since you would have a drain location at both ends of that 50-foot run, a peak anywhere along it would be acceptable provided you maintain the required slope to each drain from the peak.
