Your link says "uniformed services," not "uniformed military services." If UA said "members of the uniformed services," I would agree with your reading. But UA has repeatedly stated and written "uniformed military personnel" which would be pointlessly redundant if by "uniformed" they just meant "military." (just like saying "uniformed military personnel in the armed services" would be pointlessly redundant).

IMO, the only logical reason for adding "uniformed" before "military" is to clarify that the policy applies to military personnel in uniform.

Also, +1 to everything fastair said above regarding the reading/interpretation and purpose of the military boarding policy.