This is a follow-on issue to JDK-8195974.
The fix for JDK-8195974 eliminated the dependency on the java.logging module from the javafx.* modules by replacing all calls to java.util.logging with calls to com.sun.javafx.logging.PlatformLogger, which is in javafx.base.
A few of the systemTests -- in tests/system/src/test*/java -- still use j.u.l. These calls could be changed to use PlatformLogger with a fairly simple enhancement to that utility class. The system tests that still use j.u.l use a custom logging Handler.
Adding logging Handler functionality should be pretty straight-forward. The easiest way to do this might be to add a log(Level, String, Object...) method, which would be called by all of the convenience methods, and then have that method call the custom log handler, if one is set.