Why is Capsaicin injection not used instead of nerve surgeries for pain?
There is a simple answer to your question: it might cause a massive inflammation-like response and it might also hurt really, really badly.

Capsaicin activates heat receptors which are not only involved in heat reception (that would cause the pain as TRPV1, the receptor, is strongly activated at temperatures higher than 43 degree Celsius - accordingly, it will feel like burning from inside) but also in sensing inflammations within the body. That means that your body will think it has to take strong and immediate action if capsaicin enters tissues directly, flush out a bunch of neuropeptides and actually might start destroying the affected tissues.