I agree that interests rates between trading partners have to somewhat match but the issue here within Europe is that Germany is so big that by default what they say goes.
So, if you have Estonia that is in a slump and needs a boost, it can go to the ECB and say, look we need to lower the interest rates but if Germany is happy with the current rates, the likelihood that Estonia gets its way is basically nil.
Right now Germany is feeling the pain and despite the inflation picking up above the 2% target rate again, the ECB has not raised the rates further, why? My hunch is because Germany can't afford it and neither does France.
A monetary union is great on paper, in reality the big fish still eats the small fish. The only difference is that the small fish can try to do something outside whereas in the union it just goes along and hope for the best.
Finally, another big issue with the Euro is that there is no fiscal union between the states. So if a state like France struggles then if a federal Europe was to happen, the states who have money would give it to France in forms of tax transfers but doing so punishes the countries that have made the reforms, that have invested, that have saved money and reduced their deficits and will only incentivized countries which have not done so to continue having deficits in the future.
How would a German politician or Austrian politician explain to his/her constituents that there is no money for new schools or hospitals but there is cash to bail out Italy or France for the nth time because these countries have refused to do what was necessary to reform their country?
In any case, since the exchange rates are no longer providing the feedback that the markets used to give by repricing the currencies that existed before, the markets now use the interest rates of the debt as proxy for their confidence in each European state. Right now, there is a 85bp difference between France and Germany and its widening as time goes on so something will have to give soon.
Either the ECB caves and lower the rates which can increase inflation or France risk triggering a Euro crisis that will be much much worse than the Greek one.
And since a lot more countries are now in the Euro compared to the 2010s, the spread of the crisis will be greater by default.