> Can a citizen who is governed by the EU meaningfully influence who is doing that governing? In particular, can a large group of citizens affect who holds power within the EU and vote out those individuals they don't want, so any individual office holder has some degree of personal accountability to the electorate?
Erm yes. A large group of citizen can apply pressure to its local government to push for change in the EU, they can also to the same via MEP elections. You just need to remember that a "large group" need to be very large to be considered important relative to the EU 450 million citizens. The UK's total population only makes up 14% of the total EU population, what gives us the right to dictate terms over the remaining 86%?
National governments pick the members of the European Commision, so if you're not happy with your European Commission representative, take it up with your national government. As for the member picked by other governments, well you wouldn't expect to have power over an MP that doesn't represent you.
> It is debatable whether even the European Parliament does
The European Parliament obviously does. The only reason why the UK keeps getting short changed by the European Parliament is because we keep electing idiots into power. Most because our national government like to pretend MEPs don't exist, thus doesn't educate people on the importance of MEP elections, then acts surprised when the European Parliament doesn't represent the UK population.
> If you don't think it matters that many of our population have little influence over who occupies Number 10
I think it does matter, I think it matter quite a bit. But I'm not convinced that the general public is the best body to make that choice directly. The whole point of have a representatives is that they have the time and resources to educate themselves on the minutiae of state, and make better decisions than the general public. Not because they're smarter or better, but because they're better informed.
My view on the EU debate in the UK has always boiled down to the fact the UK public has simply not bothered to engage with the democratic systems in the EU, so it's not a surprise that those systems don't represent us. The fix here was always for the UK to actually participate in the EU, not just strope, but that would require the UK national government to stop using the EU as it scapegoat for its own domestic failures. At least with Brexit the UK government won't be able to blame the EU for everything anymore, and we might actually get some competent leaders with a real vision for the UK.