It then seems reasonable to store the checks in the same place one would store a grand of cash.

(That's where I keep my checkbook; I only just keep a single, blank, folded-up check in my wallet for the unexpected need.)

Actually with the way bank regulations around check fraud works, timely knowing (/reporting) that your physical checks have gone missing is important, so you do not want to hide them out of the way. Despite being just pieces of paper, there are different regulations depending on if a fraudulent check was created from one of your bona fide checks, or whether it was forged from whole cloth.