I discovered another site that doesn't work well with Safari -
https://www.blueshieldca.com. I've always used Safari there and thought they had a bug on their site; I couldn't use the dropdown at the top of the page to switch between my different policies. Two days ago I tried using Firefox and all worked correctly.
Blueshield should have tested against Safari. I suspect they're doing something non-standard and only tested against more common browsers. These days "more common browsers" just means Chrome variants.