Can one still become a Qari, if ones teeth are positioned in a way in which not all characteristics of the letters can be pronounced correctly?
It depends on how much it affects the letters and in which way. If only one letter or one characteristic cannot be pronounced correctly, and the student of the Qur'an wishing to get an ijaazah can correct others and describe the mistake in a correct manner, they will not be prevented in general, but it will most likely be written on the ijaazah that they read correctly except have a mistake in this letter.
If it affects several letters or several characteristics, the matter is more serious and the sheikh in charge has to make a decision about the student.
Having said that, we will say we have seen several students start out with articulation point mistakes due to a problem in the teeth, but with Allah's help, dua', great effort, and time, they were able to correct to an acceptable level. For example, students with a big overbite have problems correctly pronouncing the letters of
, which are:
; but if they practice and bring the lower jaw forward to align the teeth, many can overcome this problem, by Allah's will.
And Allah knows best.