interest rates in the in the PIIGS countries should rise to meet investor's default risk assessment

What interest rates?

The European Central Bank sets a single base rate for all of the EU.

However, you are right that there are credit spreads between the different Eurozone sovereign debt issues, as well as between firms in different sectors and countries.

It is possible that, because of monetarism, too much importance is attached to the single base rate set by the ECB where it is the other market-set interest rates that matter.

Meaning, eventually, a flow of funds to those countries from lower interest rate countries within the euro

Here I prefer to think of higher interest rates as penaties that borrowers have to pay, not as something that makes lending more attractive to them because they are offering to pay a higher interest rate. A higher interest rate spread doesn't help.

This intimates macroeconomic 'convergence' within the euro countries and, possibly, the EU as a whole.

I think the outcome of interest rate differentials is that, say, German firms, will be able to take over, say, Spanish films, because of their loewr credit spreads. I don't think that has much to do with macroeconomic convergence.

I think the macroeconomic convergence must come from local inflation differentials which cannot be compensated by the various Eurosystem central banks by means of different local "base rates".

En un viejo país ineficiente, algo así como España entre dos guerras civiles, poseer una casa y poca hacienda y memoria ninguna. -- Gil de Biedma