The FBI could have run the name through state DMV records and found everyone with that name. Depending on whether the NCIS check was retained, there would have been a match with a firearms purchase.
Given the name, I doubt there would have been many doors to knock on to determine if any had a history of mental issues.
All the FBI has to do is to run the serial number of the rifle with the manufacturer, who will report to which distributor they sold the firearm.
Then the FBI gives the serial number to the distributor who will report the name of the retailer that they sold the rifle to.
Then the FBI simply goes to the retailer with the serial number of the rifle, and the FFL dealer will produce the Form 4473 that was filled out at the time of the sale of the gun. (FFLs are required to retain all 4473s for the duration of their time in business, then to forward them to the ATF if they cease to be a business.)
On that 4473 will be a unique NICS approval number, proving that the individual on the 4473 passed the NICS check, the name of the individual who purchased the firearm, and what ID was used to identify the individual.
There is a process that is deliberately difficult enough that it can be used to trace an individual weapon used in a crime back to its owner, yet not able be used to trace all weapons owned by an individual.
At least on a federal level. States like California, New York, New Jersey, and Illinois maintain their own firearms ownership lists. Other states like Florida do not.