I'm sorry but this is simply not true. It matters whether the teacher is a good teacher and knows their stuff and not whether they are native speakers. It's good to have but it's neither crucial, nor doesn't "dramatically improve" anything. I am from an English Language School myself and I had some really great, great teachers in English - all of them Bulgarians. I had also some science classes in English too - these teachers were pretty good too. As for the native speakers - through the years I have been exposed to a couple of them and mostly - meh ... especially the American who was supposed to teach us Geography. I had a great English teacher in Media and she did her job well - she knew her stuff and was very engaging. But still this is just one class a week (in only one school year) and doesn't have substantial effect on anything. The "core" English language teachers are with you every day for at least two classes in a row. 
And lastly, nowadays children and Bulgarians as a whole are much more exposed to native speakers so that's not something really missing from everyday's life.