The difference is in the function of the verb "could", not in the verbs it is paired with.
Using "could" to say someone managed to do something is using it as a modal verb. This use as a modal verb expresses the probability (chance of something happening).
Compare this to the usage of "could" in your other example sentences.
Suddenly I could hear a loud noise.
I could smell something burning.
In theses sentences, "I could" is simply being used as the past tense of "I can."
use of "be able" to specify the specific past occasion like "He was able to speak French to the policeman to explain what had happened". Use of could in this context is inappropriate
Using "was able to" in that sentence is valid. However, I would not say that "could" is inappropriate for that sentence. Consider for example if I changed the sentence slightly to this:
He could speak French fluently enough to explain to the policeman what had happened.
This is perfectly valid as well.