Meat is not "very regular" in the modern mediterranean diet. This site ranks countries by meat consumption (overall and not per capita and given with reservations because I don't know where the data comes from, but still):
https://worldpopulationreview.com/country-rankings/meat-cons...
Ranking of the three countries whose cuisines inspired the mediterranean diet, according to the site:
Italy 24
Spain 31
Greece 87
This ranking doesn't say anything about what kinds of meat are consumed and this is culturally determined also. For instance, counting per-capita consumption, Greeks, Spanish and Italians eat more pigmeat than other kinds of meat:
https://ourworldindata.org/grapher/per-capita-meat-consumpti...
https://ourworldindata.org/grapher/per-capita-meat-consumpti...
https://ourworldindata.org/grapher/per-capita-meat-consumpti...
And all three countries raise relatively few cattle (with Greece having less than a million):
https://ourworldindata.org/grapher/cattle-livestock-count-he...
I'm bringing this up because, to my knowledge, the traditional diet in Greece, while predominantly lacto- ovo- pescaterian also included cured pigmeats as a common food item, while lamb was eaten only rarely, at religious feasts (particularly Easter Sunday and the religious feast of a town's patron saint) and cattle were raised primarily for dairy. Similar in Italy and Spain.
So the "Mediterranean diet" as it is understood today is only "inspired" by the dietary habis of the traditional diets of Italians, Spanish and Greeks. In truth, these people traditionally consumed meat in larger quantities than assumed by the modern "Mediterranean diet". Accordingly, the amount of meat that modern Italians, Spanish and Greeks eat has not changed as much, from their historical past, than is suggested by comparison with the modern "Mediterranean diet".
That said, those three peoples are still far behind other developed countries in their consumption of meat today, especially beef and particularly so Greece.
You're right about refined flour. In Italy, Spain and Greece, bread is always on the table and it is mainly white bread. Pies like spanakopita and pasticcio are also today made mainly with refined flour. Sugar is also much more widely used today, as is salt, compared to the past, because of higher availability overall.