In the video, Ben Criger shows how the current has a significant bump before arriving at the place where there is a state-dependent current. Do you think that the height of this first peak (which is independent of the spin state) matters? Why does it (not) matter?
No, in principle this doesn't matter, as long as there is a detectable difference in area under the whole curve. However, we must note that if the first peak is extremely high compared to the second peak, you could be limited by the measurement resolution. In general, the higher the signal you want to measure (or the broader the range), the lower your measurement resolution becomes. If the difference in area of the spin up and spin down states becomes smaller than the measurement resolution, you cannot distinguish the two states any more with the measurement.