In addition I have noticed that DesignGraphics that do show up on the map have a BLOB in the Element column. My additions have NULL in the Element column. If this is the issue, the question then would be, what am I supposed to put in the Element column?
We found the answer ourselves. Once you have the TextElement that looks like you want it (text, font, position, angle, etc.) you just need these lines of code:
var feature = featureClass.CreateFeature();
var annotationFeature = (IAnnotationFeature) feature;
annotationFeature.Annotation = (IElement) textElement.ArcTextElement;
Where featureClass is SDEADMIN.DESIGNGRAPHICS and you are in an editable workspace. Apparently setting Annotation fills in the Element BLOB and the rest of the columns automatically.
Most importantly, it works.