> those major language groups (English, French, Germanic, Romance, Slav)
The problem with picking a language group (English is a Germanic language and French is a Romance language) is that even native-speakers within those groups can't understand each other, let alone speak and write the other languages.
I speak two Germanic languages, but I can't understand German, and I speak one and a half Romance languages but I can't understand Italian.
> then every country can translate into their own specific language
Who pays? If the country pays, then it's unfair because very little will be published in, say, Dutch, so the Dutch speakers (who form 1.5 medium-sized countries) will have to pay higher translation costs than the English- or German-speakers (who form much larger economies and could better afford such costs).
Or should the EU pay? In that case you've just recreated the current situation whereby everything is translated into every language and the EU pays.
There is no simple answer: language policy will always provide oodles of fun :-)