Maybe they meant that people arriving in an ambulance were generally seen earlier cos they had more urgent cases on a whole.
I do believe the triage system applies to all and how you arrive has no bearing on it, (I've worked in hospitals all my working life)
I've also never heard of emergency departments not allowing people to arrive on foot, or car. I've never called an ambulance and always driven myself or others in. There's an ambulance shortage where I live and it's only 10minutes drive to the hospital from where I live, would be longer to wait.