Just to be clear, Tony Fauci is not synonymous with NIH and Tony Fauci does not personally dispense extramural research grants. In fact, he has nothing to do with it. Fauci is Director of the National Institute for Allergy and Infectious Disease. He runs that institute only, which is one of at least 10 different institutes that constitute NIH, albeit NIAID has one of the biggest but not the biggest budget of any of the institutes. He also runs a large and very productive lab (I am sure he has co-authored more than 1000 papers in peer-reviewed journals), and he is ultimately in charge of all the patients admitted to the NIH Clinical Center under the NIAID aegis. Plus, he has become a TV personality, thanks to DJT and the pandemic. I don’t know his work schedule these days, but a decade ago it was common knowledge that he typically put in 12-14 hours in his office, not to mention time spent reading at home. I think that’s enough work and responsibility for any man. NIAID specifically, and each of the other NIH institutes, have large groups of reviewers, who give out extramural grant money. (The NIAID grant review group mainly funds research related to its mission, to combat pathogens like HIV, tuberculosis, malaria-related applications.) I served on the NIAID extramural grant review committees more than once during Fauci’s term as Director; he is nowhere in sight during that process. That myth about "gain of function" is BS meant to discredit Fauci that has been circulating since before the election; it makes no sense to anyone who knows anything. Truthfully, that whole business makes me sick.
My thoughts are the virus is NOT man-made. If it were man-made, it could only have been man-made in the sense that someone took pieces of one or more other Coronavirus genomes and spliced them together. This could be done in a lab, but guess what... No one could concoct such a virus and know in advance what it was going to do. We are just not that good. Hollywood exaggerates. There is not even any lab method to predict the particular pathogenicity of this particular virus, in advance of releasing it into the human population. Further, the Coronaviruses have the capacity to re-combine in nature, which is to say when or if two coronaviruses infect the same cells in an animal, the genomes of the two separate viruses can exchange pieces with each other. These exchanges of genetic information occur at certain known points in the genome; it’s tightly controlled. The resulting chimeric viruses could/can easily be detected by genome sequencing and comparing results to the parent viruses. There are some papers that show a plausible origin of SARS-CoV2 in relation to certain native bat viruses. I don't know if any particular scenario is yet proven. In general, for any virus, the vast majority of deliberate mutations introduced into the genome by scientists in a lab result in a crippled or attenuated virus that has actually lost some of its virulence properties. It’s much easier to do that than it is to create a Franken-virus that will eat your face. There is absolutely nothing about this virus that makes me think it came out of a lab. (However, it could have come out of a lab by accident if there were a lab that was collecting Coronavirus isolates from bats or other animal hosts for a legitimate scientific purpose, but there is no way it was made in a lab.)
As to your question about mutation. Coronaviruses actually have a lower mutation rate than most viruses, owing to the fact that the virus genome codes for a "proof-reading" protein that eliminates most mutations before transcription is completed. IF there were a man-made "synthetic" Coronavirus, it would still have to obey all the rules by which Coronaviruses operate and proliferate, and I would expect it to mutate at a rate no different from any other Coronavirus. As far as COVID is concerned, the virus is constrained from mutating in such a way as to negatively affect its binding affinity for its cell receptor, ACE2. ACE2 is the receptor in both humans and susceptible animal species. This is good for us, as our vaccines are targeted to produce antibodies to the part of the virus that binds ACE2; the virus will have a hard time escaping antibodies induced by the present vaccines, although we may have that problem in future. Fortunately, it’s very easy to combat an escape mutant when it occurs, using the latest technology.