Updating the .desktop file to look like this appears to resolve this issue:
$ cat /usr/share/applications/insync.desktop
[Desktop Entry]
Version=1.0
Type=Application
Name=Insync
GenericName=Insync
Comment=Launch Insync
Icon=insync
Categories=Network;
Exec=env QT_QPA_PLATFORM=xcb insync start
TryExec=insync
Terminal=false
So something like:
sudo vim /usr/share/applications/insync.desktop
Then add env QT_QPA_PLATFORM=xcb before insync start on the Exec line, save and exit, wait a few seconds for Gnome to see the changes, stop and then restart insync.