The saying that "Don't teach Dharma to those who can't appreciate it or aren't interested." would seem to stem from the lack of belief of a person in the Buddhist teachings.
When a person is able to appreciate, or willing to believe in Buddhist teachings, then the conditions of that person to learn the Dharma have matured. That's why there is a saying that "the Buddha is unable to help those who have no affinity with him".
What is affinity? It is the ability of a person being able to believe. Even a Buddha cannot help someone whose conditions have not yet matured. However, when they have matured, the person will have belief. Then the Dharma can help and can be taught.
Here we speak of belief, that is, first, we believe in ourselves. This is where Buddhism differs from religion. In religion the most important criteria is to believe in God. In Buddhism the most important criteria is to believe in ourselves, not something outside of ourselves. Next we need to believe that we have the same Buddha nature within ourselves. Believe that originally we were Buddhas. Believe that we are no different from the Buddhas. Believe that our true nature has become polluted and that once we remove this pollution we will uncover our true self-nature.
Once we have the belief, we then are matured to seek for the meaning of Buddhism (understanding). Buddhist teachings explained to all beings the truth of life and the secrets of the universe. When we acquire a true understanding of it we can begin our practice. Practice is based on understanding. True practice is based on the foundation of principles and correct methods.
The ultimate goal of practice is to achieve attainment, to attain the real benefit of Enlightenment along the path of Buddhahood.
For undertaking the study of Buddhist teachings, having the Order of Learning is very important. That is to first establish the belief and then seek for a correct understanding before starting our cultivation and practice. Without true belief (or appreciation or interest), it will be hard to teach the Dharma.