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Mexico has no single national cuisine; it has distinct regional cuisines shaped by local ingredients, climate, and indigenous and colonial history, so a Yucatecan dish and a Oaxacan dish can share almost nothing. Treating 'Mexican food' as one thing, especially as the Tex-Mex combination plate, erases the specific techniques and flavors that make each region's cooking what it is.
A puréed chile or tomato sauce is poured into very hot fat and cooked hard, stirring, until it thickens, darkens, and reduces. This step, often called sazonar, drives off raw water, caramelizes sugars, and melds the ground ingredients into a unified sauce. Skipping it leaves the sauce thin, raw-tasting, and disjointed.
Dried and fresh chiles are not interchangeable sources of heat. Each one, such as the ancho, mulato, pasilla, or guajillo, has its own sweetness, bitterness, fruitiness, and color, so a substitution changes the dish itself. Preparation such as lightly toasting, soaking, and removing seeds and veins controls which qualities come through, and scorching a chile turns the whole sauce bitter.