JSweet is not meant to obey JLS because JSweet is "just" Java syntax. It is not Java anymore. However, some specific transformations (very few) have been implemented so that Java programmers are not confused by JavaScript runtime behavior (as explained here for instance: http://www.jsweet.org/language-specifications/#Variable_scop...).
JSweet is not meant to be at all compatible with Java legacy APIs. With JSweet, you will not even be able to access the Java Object API nor the Java String API. Instead you directly access the Object and String JS APIs. So who cares about JLS semantics then? What JSweet is trying to compile is JavaScript applications, not Java, and the examples show that it works for not-that-trival code.
In short, JSweet is for writing real JavaScript applications in a Java compile-time environment (not runtime). Nothing more, nothing less. I understand that GWT new generation it aiming somewhat at integrating JS apis too... But AFAIK, JSweet is the first project to give access to hundreds of JavaScript libraries in Java thanks to the TypeScript to Java API translator.