From the page as listed, any purchased item from Apple that was registered
with a serial number will appear; then the serial number could be used to set
an appointment (along with Apple ID) for the product.
Did you click on the image/icon of the product, & not on the link to coverage?
When I look at that page, my purchased products include models of Mac
and AirPort, portable mac, and others, from over 10 years ago. You don't?
And I can still get support/service options on my G4 iBook 12" model.
Just now. Others that I did not buy new, can also have arrangement
made for the Genius to see, diagnose, and offer advice on their issues.
The Genius people in an Apple Store have been advising product owners
on matters regarding service options, test diagnostic, and the like, even
though the products may be past their warranty &/or AppleCare coverage.
So that is why I mention the page where content is visible when you log-in.
Since it is relevant to most users who may need to know this, even after
the initial complementary warranty, a record; & remains helpful later on.
Anyway, you should be able to contact a Store or Specialist, through the
locator links, and ask they how they can contact you when a reservation
or appointment can be made concerning a specific product issue. They
can call you when a time is open, even if you don't see one available.
I was able to go online and could have set a Genius appointment at the
Apple Store in AK (not just a reseller, the official one) and could set one
for any of those older Apple products I've owned for years, as well as
the last Used (2005) Mac that I was given -- none of mine have a warranty.
In any event... your experience may vary, as do mine.
Good luck & happy computing! 🙂