Definitely not generally based on appearance.

What you suggest is not what happened at the Saudi embassy in DC in the years where I was well connected with Saudi diplomats in the area and reasonably well connected with those running the Saudi intelligence establishment -- rather recent years at that. If there is any evidence of a change in the past couple of years, it probably would be a surprise even to the current Saudi ambassador to the US.

There are many millions of ethnic European-looking muslims, and at least tens of thousands of ethnic European-American muslims living in the US. Thousands of such persons apply for Saudi visas, many of them with names that most Americans and Canadians and ordinary Saudis wouldn't recognize as being names of muslims. Yet there generally is no religious identity adjustment to their applications either.

I'm no fan of Saudi Arabia -- and I put the Saudi government in the same category as the den of snakes that is the the Pakistani government -- but at least things are actually rather obvious with their reactionary ways that making up stuff to criticize the Saudi government is just a waste since there are various legitimate grounds upon which to criticize Saudi Arabia without imagining things.