DaMoshe –
If the shul had a rule disallowing a baas tefila wearing a colored shirt on Shabbos from davening for the amud, would you have been as offended as you were in the above story?
If not, why not? If it’s because it is acceptable to classify one shirt over another as more respectable then why cannot the same concept apply to a yarmulka?
Is it, perhaps, possible that the shul’s rule is to disallow a mispallel who lacks a hat be the baal tefila but your cousin inarticulately expressed it the way he did since your cousin equated kippa sruga with lacking a hat?