Correct. However, when a background check is performed for a clearance, the same databases are searched looking for the same sort of flags that would trigger a NICS denial.
Which is how a background check for a clearance is done as well--at least that's a small part of it. There's a more in-depth investigation as well, but that's where it starts.
No, it involves a much more thorough check than a NICS performs. Which is why anyone with a security clearance should be able to pass a NICS.
The President gets a security clearance by virtue of his position. Most folks with security clearances got one because they had no adverse information appear in the process of a background check.
If a person with a security clearance is denied in a NICS check it's almost certainly not because they actually deserve a denial, it's because there's a mistake somewhere.
Of course you are right in the sense that it's always possible to be denied. My point wasn't that someone with a security clearance is guaranteed to be approved. The system isn't perfect and it can clearly deny someone even when they're not really a prohibited person.
My point was that if someone who has a clearance (and has therefore passed a more rigorous check than the NICS imposes) is denied, it's almost certainly NOT because they are really a prohibited person. The denial is therefore almost certainly invalid--that is, it is a denial issued to someone who should not have been denied. Whatever reason is given for/generated the denial is not a valid reason.