Is complete dominance actually a genotypic process?
"Or are all alleles actually codominant at the transcription/protein level?"

Yes this is the right idea. Generally all alleles are transcribed independently (with some exceptions like in females where one X chromosome is inactivated). And the dominant and recessive phenotypes are in a sense extreme cases of codominance. The terms are used for describing phenotypes and depend on thresholds of activity. In the snapdragons example we can associate 2 functional enzymes EE with the red phenotype and white (lack of red pigment) associated with 2 nonfunctional enzymes ee. But one E is not enough to make the red pigment so we see some red pigment and a pink phenotype. 

Now consider a yeast cell with 2 hexokinase genes (H). With two inactive proteins (h/h), the cell will grow slowly. Now let's say that H is a very efficient protein and that glucose transport into the cell is rate limiting so one active H has the same phenotype as having 2 H/H, so H/H is the same as H/h thus H is a dominant allele and h is recessive because one H is past the threshold of activity for fast growth on glucose. h is recessive because h/h genotype is needed to see the h phenotype. Hope that helps.