Moving from hot to cold wallets in an environment with redundancy, multiple partial keyholders, and offsite storage could easily take days to weeks and require scheduling access to the physical vault holding the cold storage. You're conflating hot and warm storage - warm can replenish hot but cold requires physical access to somewhere ideally unrelated to where day-to-day operations run.

Xapo, one of the biggest cold wallet firms in the world, has a security feature that you can request where their people fly on a private jet to meet you personally and confirm face to face that you indeed are the one making the request to move funds from their cold wallet.