If you look here https://covidtracker.fr/vaccintracker/ you'll see that on the second graph, the blue curves (vaccination) closely follow the gray curve (deliveries).
And while France is one of the most anti-vaccine countries in the world, there is still no shortage of people wanting to be vaccinated.
In the end, I just think it is just the result of negotiations. Now, it is essentially a zero-sum game, the only way to get more doses is to take it from others, and the EU wasn't as aggressive. We will all get vaccines in the end, it is just about how much you value your head start.
Now, as I've said, there's demand\need since the EU scaled up the vaccination efforts however, look back 3-5 months.
EU was discussing just how to split\manage the vaccines among the union members, while Israel was vaccinating at a neck-breaking pace.
While my family in EU were not sure if they want to be vaccinated, people in Israel were waiting hours in line to even get a chance to get vaccinated.
Israel currently has over 80% of the population vaccinated, I'm willing to bet that even if you provide that number of vaccines to any European country, the numbers will be far lower.
I think now EU is far more active when it comes to the efforts but initially, they were not.
It seems like there is more resistance in the EU to vaccination and data sharing than in Israel.