Limited numbers of non-white soldiers are listed as such in Confederate records at enlistment, descriptive rolls and POW records; however, you are correct, many of the full and mixed-blood non-whites are only discernible due to family information and a cross-referencing of names with other sources, such as tribal rolls, census data, etc.
In the spring of 1862, the majority of the whites and "inter-married" whites were removed from the Native-American units in the Indian Territory. Unfortunately, many whites, especially from Western Arkansas were joining Native-American units to avoid more "frontline" service or being transferred east of the Mississippi River. This removal was deemed necessary to prevent this problem.
One of my white ancestors; however, was from Polk County, Arkansas and originally a member of the 4th Arkansas Infantry. For reasons of age, according to his service record, he was dropped from the rolls prior to the unit moving east of the Mississippi. Out of patriotism or maybe "for love" (he later married a mixed-blood Cherokee woman) he joined the 2nd Cherokee Mounted Volunteers (Adair) (CS) and apparently continued through the end of the war. So, he was a member of a Native-American unit, though a "white", so just as there were unidentified minorities in Confederate units, there were also unidentified whites in Native-American units. One of my interests in the study of the Civil War, is the oddities and paradoxes. Making any kind of absolute statement regarding participants in the war, their views, their reasoning, etc. is simply not possible, as this little story makes clear.
Pam, I didn't understand the "meaning" of the first paragraph in your posting. Could you please elaborate on the thought you're trying to convey and how it relates to the "loyal" Black Confederates statement?