I have just been trying to do the exact same thing on ArcGIS Pro (v 2.3.3) and got NULL for everything. I repeated the same process, using the same data, on ArcMap (v 10.5) and it worked fine. Therefore, I'm wondering if there is a bug with the Spatial Join geoprocessing tool on Pro when using 'Closest' as the match option.
Have you tried increasing your search radius? It may be the case you get NULL's for some when there is no feature within the search radius to join to...