The Hanna checkers like the Nitrate, Nitrite, and the Ammonia are very similar to the Liquid tests in that the base information comes from a chemical reaction that changes the color of the solution. The only difference is that the Hanna checkers provide a light source and photocell to measure the color change. This makes the test more precise (ie a finer scale) but not necessarily more accurate (ie does 10 ppm of NH3 from the instrument mean there really is 10 ppm in the water). The Hanna checkers are still dependent on the quality, age, and use of the the reagents to get good results, what they do is remove the subjectivity of a human observer for the color.
The checkers are likely calibrated before leaving the factory but unless they have a pre-test phase where untreated aquarium water is used initially to reference the checker before it is mixed with the reagents it will not be super accurate. For example aquarium water with high amounts of tannic acid has a chance of throwing off the colorimeter in the unit, whereas a human observer may be able to mentally adjust to the starting color of the water. I don't have one of these testers but I would not trust them unless they have some sort of referencing process.
It would seem to me that in both methods the Hanna Checker vs the Liquid tests you still have all the issues about keeping and mixing reagents to generate the indicative color change in the water. For a typical aquarium keeper's needs we don't need the precision that the colorimeter provides. In particular with Ammonia and Nitrite we are looking at not seeing any measurable amounts, both these tests are pretty clear and easy with the API tests. I don't see where the additional cost of owning and using these checkers would help with any decision I would make with the maintenance of the tanks.
It should be noted that the pH and TDS checkers use a different process dependent on the conductivity of the water under special circumstances and are quite different from the process used by the liquid tests.
