No. Swing does not suffer from that problem because Sun actually cared about accessibility requirements back in the days, in fact, the only reason there's a desktop on Linux that has decent accessibility, Gnome (through the Orca screen reader and gtk accessility libs) comes from Sun's interest in having an accessible desktop for Solaris. KDE's track record is miserable in comparison to Gnome. Sun provided much of the funding for the initial efforts in making an accessible desktop.

Qt is partially accessible, in practice, it doesn't work well.