Faith is accepted in Vedic philosophy also in the sense that you have to have some faith to read a book, some faith to approach a teacher, some faith to try a path. If there is no faith then nothing would be done. But “faith” is not the right word for this. The word is sraddha. This word has no English translation. It means reverence and trust and affection. And even this trust and reverence is not opposed to reason and observation. That’s why there is so much philosophy to understand rationally, and there is a practice for practical realization, step by step, in this life. This type of practical realization and philosophical discussion is not there in any other system in the world. Even modern science is a closed system in the sense that it operates under certain assumptions. You can only improve things within those assumptions, but you cannot challenge those assumptions. But in Vedic philosophy, you can challenge anything, and we try to answer everything. If we cannot answer it, then it is a sign that our understanding is not yet perfect, but the question is not invalid.
Regarding the Jungian style archetypes, they are called mahattattva in Sankhya philosophy. You can call them “essences of greatness”. This mahattattva exists in every person, and it exists as your personal ideals. For example, some people might think that hard work is an ideal, honesty is an ideal, etc. And others might think that honesty and hard work are for the weak; ultimately might is right. So, there are many different ideals, and when the soul enters the material world he accepts some ideals, and based on those ideals the rest of the material life is shaped. So, there is no need to reject Jungian-style thinking; but we can reject the idea that soul and God are mahattatva. Western psychology doesn’t know how to go deeper than moral principles and ideals of life. So they claim that soul and God are nothing but these ideals. But for us, these are material principles, and soul and God are transcendent to matter. Hence, religion is also transcendent to these moral ideals.
All these things are discussed in different books. You should try to read them one by one.