It could also be that these other soldiers do have a bond with a spren and are simply in training to become knights radiant. Note the scene in WoK where Kaladin says the second ideal.
Just a few paragraphs later,
This, at least to me, seems to suggest that the ideals of the knights radiant and their orders weren't something taught they were something that one found for themselves. I would argue that these other soldiers had nahel bonds but hadn't found all of the ideals yet, thus they glowed much more faintly than the full radiant.
EDIT: the above quotes are from chapter 67