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.
A lot of safety related legislation and regulation seems made with the middle class in mind, ignoring that the poor exist.