Actually, in many cases, calling directly doesn't bypass the fax system because the staff is often instructed to receive information via fax for record-keeping. For instance, medical offices and legal firms still require formal fax submissions for documents due to compliance reasons. You might reach a person, but they'll likely tell you to send the fax anyway. It's frustrating, but that's how some systems are set up.
Yeah, I ran into the same thing trying to book a campsite last year. They only took mailed paper forms and a check. It feels weirdly charming until you realize you might not get a spot.