Hearing and vision are both very sensitive senses, that are easy to permanently damage.
Everyone should have access to treatment they need, but having a licensed doctor review severe cases to find the right fit and prevent further degradation or full deafness, especially if their hearing is already weak, is a reasonable safeguard.
You can use an app to check which frequencies are missing. The man with the bit of paper will be doing the same thing.
I may be biased as I work on a medical device and at least for the device I work on the doctors signing off are contributing nothing more than their license.
But also it's protection to prevent you from further damaging your hearing, just like most drugs require a prescription.
The real problem isn't the regulation requiring professional oversight of potentially damaging devices (or medications), but the fact that medical care is affordable for many.
Everyone should have access to treatment they need, but having a licensed doctor review severe cases to find the right fit and prevent further degradation or full deafness, especially if their hearing is already weak, is a reasonable safeguard.