Are there people out there who can provide first hand account or refer me to literature that discusses the influence of PhD advisor on your chances of getting a faculty job?
That is a very good question and more complicated to answer because of different dynamics which can occur during your PhD. I want to keep them short, but I have first- or second-hand experience in all of them:
Influential supervisor: Is Lead-PI; has several projects; has some PhD and Pos-doc students; has a high h-index; etc.
- You have an influential supervisor and you come along with him during your PhD: You will have high chances to get into a good position; the recommendation letter will look great, and you will have done a good job publishing high-impact papers.
- You have an influential supervisor and you both get not really along: You need to count one or two extra years for your PhD; your publication record will be ok (generally, also depending on the requirements of the university); your recommendation letter will be more or less. From my experience, people in this situation always go to industry even though I think they are good scientists,but they just got disappointed.
- You don't have an influential supervisor and you both don't get along with each other: Disaster!
My recommendation: It is more important that you have a good working atmosphere with your supervisor than to have a supervisor which is involved in too many things and he/she will not have time for you.