There is a V sound in some Chinese dialects, such as Hakka. If you listen to northern Chinese speak, sometimes they use a V sound.
But as to why there is no V sound in Mandarin, first, languages can be pretty arbitrary. English has words beginning with shr, such as shrapnel, shrink, shroud, and others, but almost none beginning with sr. Why not? Because. Taiwanese languages have lots of words beginning with ng, such as ngahi (yam) or ngungu (fear) in Tayal, nghou (monkey) or ngohza (cherish) in Tsou, or ngayaw (go headhunting) or ngiuwngiuw (gripe) in Amis, but English has no words beginning with ng. But most Taiwanese langauges do not have T or P sounds. Tayal also has no F sound.
Also, v/f apparently entered languages quite late. It has something to do with agricultural diets requiring less powerful jaws, which permitted the v/f sound to develop. That began in Chinese only about a thousand years ago, if I remember correctly.
But asking why such and such a language does not have such and such a sound is ultimately futile. the answer is, Because.
The Ability to Pronounce ‘F’ and ‘V’ Sounds Might Have Evolved Along With Diet