According to the European Commission's statistics, it employs 32000 people -just over 1000 of whom are detached / seconded national experts (from all 27 member states).So the secondees are actually only a very small part of the EU staff. They can also only stay in Brussels for four years before having to return to their own government. They tend to get the more technical posts dealing with policy implementation, based on their previous national roles, so are concentrated in certain directorate-generals like competition and transport and I think are also often found in the executive agencies (but if you want policy work that's not so interesting). So if you want to make a career there (and get the more interesting posts) then there's nothing for it but to try and get through the concours, which incidentally Member State governments are not allowed to influence (to avoid obvious nepotism).
If anyone is really keen on an EU career and has a politics/ economics/ law degree and speaks French, it's worth looking into the College of Europe MA programmes. The UK government offers a certain number of scholarships each year and you'd certainly learn the material necessary to have a good chance of getting through the concours, and it does give you the chance to build up enviable networks of contacts in Brussels so can help getting a Brussels-based job in political lobbying, consultancy, NGOs etc as well as the institutions themselves.