The point is to create inconvenience for any employee who claims x. You might as well ask the employee to fetch a stone from the top of the nearest mountain.
It might just a CYA move on the part of an individual actor in middle management. The manager doesn't want to risk getting penalized for handing out "random perks", but also doesn't want to get in trouble for ignoring a "medically necessary" request.
Put another way: That seemingly arbitrary behavior might make more sense if you assume that it's intended to protect an individual, rather than to protect the corporation.
The negativity around giving "random perks" makes it feel like you're running a prison, where the point is to punish people instead of a workplace.
Sounds like the employee in question was trying to get special accommodations they didn't really need and their employer was wise to their bullshit games. OP is the kind of person who makes these ridiculous authoritarian gatekeeper rules necessary.
... No. Your employee who uses a computer shows up and says they are blind and need Braille accomodations? You don't wait for a doctor's note, that is explicitly unreasonable.
Gatekeeping is appropriate, if at all, where an accommodation is nice for lots of people who will somehow cost the company a lot of money if they receive it and it is difficult to determine the reasonability of the request on an individual basis without invading their privacy - so you ask a doctor to stand in as a firewalling proxy that tells you whether it is reasonable in this case without telling you the underlying cause.