Nowhere have I stated that asubha is the core teaching of the Buddha - either according to texts or according to practice.
Asubha is just one mode of kāyagatāsati (others being mindfulness of breathing, four elements, postures etc.). Buddha stressed the importance of practicing kāyagatāsati for the realization of the deathless. This makes a lot of sense is you think about it as there is so much attachment, delusion & wrong view connected with the body.