"My mistake. I was thinking of the European Single Market, which Switzerland does opt into."
Indeed. And the two animals (the Customs Union and the Single Market) are completely different animals. Interestingly on that topic, it is said we must engage with a customs union to prevent a hard border in Ireland, where the cross-border trade is (relatively speaking) two pennyworth of nothing. However, no such problems seem to exist in Switzerland which has no hard border with any of its four EU neighbours. But once again that's not something I'd know much about as I voted to leave the EU.