Here my note on not generalising these anecdotes should apply: I don't know how important a factor Russian language instruction was overall. My relative is the only of two sources I read emphasizing this: it wasn't among the students' 16-pont demand, I only saw it mentioned as one of Kádár's initial offers when trying to gain power. On the other hand, I take the main reason behind the rejection was that it was compulsory yet of no practical benefit for most.

we have a stereotype of Hungarians as all speaking half a dozen languages impeccably

That's because no one can talk Hungarian :-)

I recently read an article that claimed an EU study found that Hungarians are near the bottom in terms of foreign language proficiency in the EU

These polls (it's a regularly repeated EU poll) were discussed before on ET, for example here. Some points to alter your interpretation of them:

1. those who really don't speak languages are mostly the rural population;
   
2. another real factor is that unlike Slavic, Germanic or Romanic languages spoken by almost all other Europeans, Hungarian has no closely related language (it is not even Indo-European), thus learning a foreign language is difficult by default;
   
3. these percentages are  poll numbers, based on voluntary self-judgement. I find it likely that more Hungarians than others who did learn foreign languages and are able to communicate at least rudimentally judge their capabilities as just not enough.