Boy, we have a lot of circular arguments going on here. First off, real world, not the court of the internet. 47 years of serious flying I have never, ever been asked to provide my certificate in the USA. Overseas, yes, nearly every time I land. Now reality, your friendly FAA inspector and/or NTSB investigator can access your records online, all of them, with the exception of medical records, aside from you did it and the application (HIPA rules). Your BFR and IPC are not online.
So, no, as you lie on the ground next to the smoldering remains of your aircraft, do you have to produce your logbooks, the aircraft's records, all your certifications and endorsements. You do have to be able to provide them outside of your physical license and medical within a reasonable period. Personally, I have a post grad degree in Aviation Safety and am a certified accident investigator and have worked on contract for NTSB and the FAA along with the insurers. So I have seen that process from the inside.
Now if you want to get the ire of the FAA, lie to them. Tell them your current and your not will result in a huge hammer fall on your head. As I mentioned, they can get it all online in seconds. The only thing they cannot see without your logbooks is your BFR or IPC. And if you lie about it...