I don't believe that is true anymore.Yes, an FBI PCC can be requested at any time. It is typically taking (now a days) around 3 months to receive the FBI PCC due to their backlog. I would recommended requesting the FBI PCC well in advance, since post ITA, you may not want to wait for too long...
For citizens, the processing time is always fast.
For people who are not citizens and not PRs of the U.S., the FBI changed the application process in 2018 or 2019. Now it is much faster.
I cannot remember the case for PR of U.S.A.