I also don't really understand where or how the consensus of Bulgarian having "no inflections like English" comes from.
If we take a look at the definitions between analytical and synthetic languages, we have the following:
Analytical language: Any language that organizes words and grammar by strict word order, particles rather than inflection to convey meaning.
Synthetic language: Any language that organizes words and grammar through inflection to convey meaning rather than relying on word order.
Now if we compare some text in English and Bulgarian:
English: Even though he had pain from arthritis, he studied hard because he wanted to go to medical school.
Bulgarian: Въпреки болежките от артрита, учеше усилено, защото искаше да отиде в медицинско училище.
We notice that болежките is an inflected form that serves a purpose in the sentence. болежки means pains and болежките means the pains (the pains that he had) the suffix те is a definite article that acts a lot like (is) an inflectional ending belonging to the noun, учеше усилено can also be усилено учеше in English this translates to only he studied hard here we notice that word order is relatively free. Also артрита is an inflection of артрит which can act either as a genitive case or a definite article (still an inflection) to describe that the arthritis is causing pain.
So if we compare the flexibility of word order of both sentences we get.
English: Even though he had pain from arthritis, he studied hard because he wanted to go to medical school.
Bulgarian: Въпреки болежките от артрита, учеше усилено, защото искаше да отиде в медицинско училище.
Bulgarian: Въпреки болежките от артрита, усилено учеше, искаше да отиде в медицинско училище.
Bulgarian: Въпреки болежките от артрита, усилено учеше, искаше в медицинско училище да отиде.
Bulgarian: Въпреки болежките от артрита, усилено учеше, искаше в медицинско училище. (Notice that да отиде is entirely optional.).
Bulgarian: Въпреки артрит болежките, усилено учеше, искаше в медицинско училище.
Bulgarian: Въпреки болежките, усилено учеше, искаше в медицинското.
(Here болежките implies/defines that he is suffering from arthritis and медицинското implies/defines medical school.).
So using this simple example and coming back to the above definition, why is Bulgarian considered "analytical"? This classification doesn't really make any sense.
Does it organize words and grammar by strict word order? No.
Does it use more particles rather than inflection to convey meaning? No.
Does it have strict word order? No.
Does it have inflections? Yes.
Does it rely more on inflection to convey meaning rather than strict word order? Yes.
As illustrated from the example above, Bulgarian clearly has more inflection and word order variation in comparison to English and is surely more synthetic than analytical by a large degree.
So, why is it classified as "analytical"?