Gluten is an allergen for people with celiac, just not IgE-mediated. Celiac shouldn't be classified with lactose intolerance, it's much more serious, arguably more dangerous then even IgE-mediated allergies.
> (digestive problems vs anaphylaxis) ... and would probably be better classified as an intolerance similar to lactose
is misinformation — "digestive problems" and "intolerance" drastically understate the seriousness of celiac disease, and contribute to the educational challenges surrounding the condition.
The chronic impact of gluten exposure to a celiac patient include: permanent damage to the small intestine, malnutrition, reduction of lifespan, and significantly increased risk of comorbid autoimmune diseases (MS, diabetes, and more) and cancer.
The acute symptoms span from gastrointestinal symptoms to neurological issues, systemic pain, and cognitive impairment. The reason for all of this is gluten causes the celiac patient's immune system to mobilize and attack the patient, i.e. "autoimmunity." Not anaphylactic, correct, but not a digestive problem nor an intolerance.