An airport is not Cat I/II/III. A specific approach is.
Every ILS array, every IFR-rated pilot and pretty much every IFR-equipped aircraft (even a GA pilot in a Cessna), can handle a Cat I approach.
Additional ground equipment is required for a Cat II/III approach, as well as additional crew training and aircraft equipment. For cost reasons, this is mostly limited to airlines and their hubs.
Assuming properly equipped and working ground equipment and aircraft, and a properly trained and current crew, when cleared for the “ILS”, airline crews will execute the Cat II/III version of the approach if weather or maintaining currency dictates.
If anything does not meet those standards, a crew can only execute the Cat I version of the approach. If the weather at the airport is worse than Cat I minima, then they have to cancel or divert—something airline management doesn’t want happening at a hub, which is why they’ll spend the money making sure that doesn’t happen there.