The fact not many launches happen from continental Europe has nothing to do with capabilities of the space industry or lack of expertise, but rather location.
Europe is far from the equator and densely populated so rocket launches from there are more costly, and failures can have a higher risk to life.
Arianne space has had 350+ successful missions after 45 years of operations, with most of those launching from overseas territories.
I don't understand the one country vs all of Europe - given a fixed population, being united in a single country surely makes everything a bit more efficient than being split in many small countries? India is much bigger than "all of Europe".
Consider India, once single country vs all of Europe: 88 orbital launches successfully reached orbit using Indian built rockets starting in 1980.
This is out of 104 individual liftoffs.
Over 46 years of successful launches.