According to my conversation with them (asking about a couple of things) - yes, the only way to file is over the phone. Why you can’t do it online is beyond me (aren’t forms to do, do, anyway?). But my hold time last week was only ~20 minutes.

CSR insurance claim can be done online (although there are some equally stupid requirements when filing, including last time, when I needed to add a made up flight to the claim, which was for a hotel (this was a road trip), even though you are not required to travel with a common carrier to get coverage, unlike with Amex).