I am always faced by this problem when teaching models of malocclusion to undergraduate students. The interest in this answer is academic. Clinically, we need to set treatment goals for both the incisors and molars. My very humble, non-evidence based opinion, is that the incisor classification is more reflective of the underlying anteroposterior jaw relation in many instances compared to the molar relations. In fact, due to rotations especially in the upper molars, the molar relation many times do not reflect the true relation of the buccal segments. There was a paper suggesting using the first premolar for classification instead of the molar relation.
Angle classification is more commonly used being easly identified despite its disadvantages.For more diagnostic precision incisor classification is recommended.
In my oppinion, the canine relationship is more important than molar relationship, because canines have more importance either in function or in esthetics in comparison with molars. Furthermore, after the emergence of the extractions in orthodontics, the molar relationship lost importance, we no longer need to reach class I relationship always. For example, sometimes we chose to extract only on the upper arch, consequently we will finish with molars in class II relationship. I use to say that if Edward H. Angle was made his classification nowadays, he would classify with canines. Comparing incisors with molars, I believe that classification of molars is more reliable because those positions are more stable than incisors. The incisors are very unstable and may undergo changes in position easily, for example in the presence of habits, therefore, are not as reliable in classifying a malocclusion.
I think demographic preference is the cause. This is like an egg/chicken question. Angle molar classification is used in USA, while BSI is used in UK. I think a better way to classify should be based on area of problem. However, classifying all three (molar, canine and incisors) helps with treatment planning rather than classifying on basis of molar only or incisor only. BSI is really relevant because crowding is known to occur in anterior segments due to forward vector of occlusal forces. I really like to classify all three teeth in each patient to give a mental picture as opposed to saying molar I and disregarding anterior occlusion or incisor II/2 and disregarding posterior occlusion.
Incisor classification is more clinically acceptable and patients know more or particular about their front teeth than the posterior because of their aesthetics