To achieve the native modal dialog look on macOS Big Sur and later we have to use the following code:

QPrinterDialog dialog;

dialog.setParent(parent, Qt::sheet);
dialog.setModality(Qt::ApplicationModal);
dialog.show();

QEventLoop nestedLoop;
QObject::connect(&dialog, &QDialog::finished, [&nestedLoop](...){ nestedLoop.exit(); });

nestedLoop.exec();

For QPrinterDialog this produces the expected outcome only if the system has any registered printers. If QPrinterInfo::availablePrinters() is empty the call to QPrinterDialog::show() results in an invisible printer dialog which still locks the application until the user presses ESC.

Using QPrinterDialog::exec() in the same constellations - no printers known to the system - opens the expected printer dialog. The problem seems to be with the implementation of qprinterdialog_mac.mm which does different things in exec() and show().