It's not really about complete vs. incomplete — it's about processing level and what comes with it.
Proteins are built from amino acids, nine of which your body can't produce and must get from food ("essential" amino acids). "Complete" proteins (most animal sources, plus soy and quinoa) contain all nine in useful amounts; most individual plant proteins are missing or low in at least one.
You don't need to combine complementary plant proteins in the same meal, as older advice once suggested — eating a reasonable variety of plant proteins (legumes, grains, nuts, seeds) across the day provides all nine essential amino acids for most people, without careful meal-by-meal planning.
The inflammatory relevance of protein isn't really about complete vs. incomplete — it's more about processing level (whole chicken breast vs. reconstituted deli meat) and what accompanies the protein (grilled fish with vegetables vs. fried, breaded protein).
This is why THE FAD's Food signal cares less about protein source and more about how minimally processed it is — a well-prepared plant or animal protein both score reasonably; a heavily processed version of either doesn't.