I don't understand how public speaking ability should correlate to my success as a Psychiatrist.
It can affect your performance if you head group threrapy-which psychaitrists usually don't do. The only times I've headed groups, I volunteered to do so, or was made to do it in residency, but that was very atypical. (I think it was the hospital just trying to save money). I do think however that you should at least do some group therapy in residency to learn about this. At least in all the states I've seen, group therapy needs to be headed by 2 people. One person often takes a more passive role. That person can just sit & nod.
It can also affect your ability to give presentations, which again is atypical. Most clinical psychiatrists do not do this. You may have to do a grand rounds at the end--a 1 time presentation at the end of residency.
I don't know to what extreme your situation is. I do know however that several people I know have the same problem & were able to do their grand rounds fine.
Another factor is several programs have a morning rounds where a resident presents a case. All IM programs I've seen have this, not all psychiatry programs have this. You possibly could be the resident who has to do this. If your program does this, it'll rotate between you & the other residents. Depending no how often that would be, and how many residents there are--it could be once every few days to once every few months.
I don't know if you'd consider this public speaking, but in treatment teams which usually number about 4-8 people (MD/DO, psychologist, Nurse manager, social worker, substance abuse counselor, a few nurses among others), the psychiatrist is expected to head this. (Then again I've seen some psychiatrists not head the team due to laziness, but they are supposed to ideally be the treatment team leader). To most this is not really public speaking since it is not speaking to a large crowd.
If the problem is on an order that you feel you cannot handle, and it may affect your performance, I suggest you do speak to the PD about it.