The assumption that things like this necessarily increase support for republicanism is probably misplaced. In Italy and Greece Governments have been effectively overthrown by the EU (or ECB/IMF) in fairly open ways with technocrats put in charge. In both countries support for the EU is running at very high levels despite that. Higher levels in fact that when it happened. The U.K. monarchy clearly lacks the wherewitha…
How, exactly? I don't recall the EU appointing technocratic governments there, i recall elections and the Greek and Italian people making a choice. Choice in specific circumstances that probably influenced them, but same as any election during a crisis.