Unfortunately I was not able to get the One to One Closest to work within a model. I was using a fairly large data set, 8k features.
I was using it in an attempt to thin features to eliminate overprint. It's a bit of a mouthful but, the model created polygon buffers for the points, ran an intersect to find areas of overprint, dissolved the polys so there was one per overprint, converted those polys to points and tried to select at a one to one relationship to remove one of the offending points.
I was able to come up with a work around using feature ID's but since I was only concerned with the one to one relationship and not necessarily the closest it might not work for your case.
-Pete