Your scenario is ideal, ron., and the euro was devised with exactly this is mind.
Now for the problem. The common currency was devised on the assumption that the economies of the members would converge, that the fiscal regulations (tax, public expenditure, etc.) would converge and become similar, that the economic output of each of the members would be similar.
Of course none of this happened. The difference in productivity (and hence the “wealth”) between Germany and Greece is absolutely enormous. The architects of the euro were warned that it might work if everything remained hunky-dory. But of course it did not. The excrement hit the air conditioning seven or eight years ago and the rest is history.
The euro is a fundamentally flawed project. Currency union requires fiscal and political union. The euromaniacs assumed that the one would force the other. It did not and it will not. There is no democratic will for political union across the Eurozone and it will not happen. To have launched a common currency without fiscal and political union was a folly of the utmost gravity and probably the most serious political blunder since WW2. It has caused enormous instability and has consigned large numbers of people in the lesser nations to penury which will take generations to recover from. The politicians simply will not admit their folly and will defend the euro to the bitter end and they are prolonging the agony of particularly the Greeks but also of some of the other “PIIGS” nations because of their vanity.