It would be cool if there was an option in Bitcoin Core for disabling the system tray entirely.

QApplication::setWindowIcon(networkStyle->getTrayAndWindowIcon());
    setWindowIcon(networkStyle->getTrayAndWindowIcon());
#else
    MacDockIconHandler::instance()->setIcon(networkStyle->getAppIcon());
#endif
    setWindowTitle(windowTitle);
#if defined(Q_OS_MAC) && QT_VERSION < 0x050000
    
    
    setUnifiedTitleAndToolBarOnMac(true);
#endif
    rpcConsole = new RPCConsole(platformStyle, 0);
    helpMessageDialog = new HelpMessageDialog(this, false);
#ifdef ENABLE_WALLET
    if(enableWallet)
    {
        
        walletFrame = new WalletFrame(platformStyle, this);
        setCentralWidget(walletFrame);
    } else
#endif 
    {
        /* When compiled without wallet or -disablewallet is provided,
         * the central widget is the rpc console.
         */
        setCentralWidget(rpcConsole);
    }
    
    setAcceptDrops(true);