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?
Well, there is the less direct answer to your question, which is that between providing funding, data, infrastructure, reaching out to colleagues, etc. your PhD advisor will have a profound amount of impact on your work, which will then have an impact on your faculty job search. For example, my project really took off when my advisor told me to email someone who had data, and to make sure to mention I was his student.
But I expect what you mean is the more direct question of how does one's advisor impact the actual job search. And, as with many things in academia, the answer is "It depends."
At the profoundly influential side of the spectrum, I have known people who have called in favors for their students and essentially conjured up jobs, or at the very least prompted a search that might have been a long time down the line. At the same time, I have also known advisors who have been reasonably hands off about the entire thing beyond writing letters, positive advice, and providing a supportive research environment. It all depends on how much they're willing to throw their weight around, how receptive their audience is to that, and whether they have favors in the right "place" (for example, if you're headed to a slightly different field, someone with a lot of clout might not be able to carry it as far).
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.