Up until the war on terror and the US no longer accepting that excuse. They do extensive KYC now.
GP's tenses are both correct. It has that reputation today, as a lingering effect of how it didn't pry historically. The industry is hanging on for the time being, probably thanks to network/incumbent effects, inertia, and some limited observance of the old ways, for clients whose countries will allow it.
there is the "are you a US citizen or a green card holder" question on every application form. probably there are two separate workflows depending on the answer.
for US citizens
Swiss banking system changed a lot during the last 15 years.