The accessible name computation seems complex with an algorithm and 17 subcases for HTML. How can I get an element's Accessible Name in javascript in the browser?
Perhaps a couple years too early for your question. The Accessibility Object Model (AOM), similar to the Document Object Model (DOM), is currently being worked on. When finished, it'll allow you to query the accessible name from javascript.
Note that your "17 subcases" are all just restatements of the accessible name computation rules. I'm not sure why the W3C has that page. For example, the first bullet point for <input type="text"> says:
Point #1 is the same as step 2B and 2C in https://www.w3.org/TR/accname-1.1/#step2
Point #2 is the same as step 2D
Point #3 is the same as step 2I
Perhaps the 17 subcases is easier to read/understand than the actual accessible name computation? Not sure. But they say the same thing.
Each browser calculates the accessible name so you really need an API into the browser to get the accessible name.