Why this order
The keyboard is first because it takes two minutes and finds the fault that most often means replacing an entire top-case assembly, which is the single most expensive thing you can discover in a hurry. The trackpad follows because on many laptops it is part of that same assembly, so the two answers combine into one repair decision.
Audio and camera sit in the middle: quick, and each rules out a component you would otherwise have to take on trust. The display tests are last because they need the room conditions to change, and doing that once at the end beats doing it twice.
What this flow cannot check
Three things matter as much as anything on the list and none of them is testable from a browser: battery health against design capacity, drive health and powered-on hours, and how the machine behaves thermally under load. Ask for the first two directly, because every operating system can report them and a seller who will not show you is telling you something. The third needs the machine running something demanding for several minutes, which is worth asking for on any laptop you plan to work on.
Also worth thirty seconds and not on this list: plug something into every port, and check the hinges and the charger. A browser has nothing to say about any of them.
Where your progress is stored
In this browser, in localStorage, and nowhere else. There is no account, no sync and no server that could receive it. Clearing it removes it completely, and you can see the entry yourself under Application in your browser's developer tools. That is the same promise the tests themselves make, and it is why this flow has no sign-in.
Reading the results
Each test tells you what it found; the guides tell you what it costs. Start with how to check a second-hand laptop before you buy it for the negotiating view, or how to check a used monitor if the display is the whole purchase.