At the very end of the "Text Products" list, there is a link to this JSON source:
This has everything we need! You can open the link in a web browser to see the JSON data:
For each storm, there's a name, a location in terms of latitude and logitude, a classification, as well as other ancillary information. And the CircuitPython PyPortal library makes grabbing and parsing this data easy.
This PDF has more information about the JSON data source: