"AFAICT, the Germans and the French are more annoyed at bailing out the south than the south is at being bailed out."
The Germans and the French were more than happy betting on Italian and Greek bonds which were junk-level and provided great yields. But, when in Greece's case the chickens came home to roost and their risk was about to materialize they forced a bailout program on the Greek people instead of allowing them to default. Having a country in the Eurozone defaulting their debt would have devastating impact on the Euro too, so there was double-whammy of incentives for the EU technocrats to bailout Greece.
After the Greek economy got devastated, mainly German companies entered the Greek market and obliterated the local competition, especially in the retailer industry with brands like Lidl coming in and undercutting the Greek businesses which had been around for decades. Moreover, they bought out a lot of those Greek businesses for a piece of bread and merged them with their own. So Greek people were essentially spending their bailout money buying products from the Germans. At the end of the day, Germany was moving money from one pocket to the other and also making interest on top of it. No EU technocrat bailed out Greece out of kindness.
Now of course, the propaganda machine will lament every day to the average Joe about how the lazy Italians and Greeks are stealing their money. So the 'annoyance' is understandable.
"AFAICT the Greeks trust the EU government a heck of a lot more than they trust their own."
False. They distrust them equally and for different reasons.
"Ask the Poles and Ukrainians about what they think about EU membership. By some metrics Poland is the fastest growing economy in the world."
I don't know Polish people and I don't know what their sentiment towards the EU is. The Ukranians have no idea what it is to be a member of the EU, they can't know from the outside. If just the fact that a country wants to join the EU is an indicator of how great the EU is, then the other way round is also true (UK, Switzerland and Norway not wanting to join, or several countries having a 'leaving' sentiment brewing within).