I've tested Java Web Start. As on Windows, it works despite Java applets no longer being allowed in browsers, as long as the developer builds in Java Web Start support.
The user needs to jump through a bunch of hoops to get this to work, but the result is a local JNLP file that launches the Java software without needing a browser.
The user needs to go to a download link for the JNLP file, and save the JNLP file despite all the weird things that happen when you try to do this in various browsers in various Macintosh environments, such as having a Discard option but no Keep option. But once the JNLP file is rescued from Downloads, for the first launch you need to right-click on it and choose Open, and answer that you are sure you want to open it. Then you need to do the usual Java security choices (as on Windows the Java code needs to be signed with a Java certificate). Afterwards you can just launch the software from the JNLP file.
Presumably if the developer double signs the Java code with both Apple and Java certificates the procedure is easier (on Windows you only need the Java certificate). So although it is much easier on Windows, it does work on Macintosh for users willing to jump through the byzantine double security procedures.