Even some really old (2000s-era) junk I found in a cupboard at work was all hot-swap drives.
But more realistically in this case, you tell the data centre "remote hands" person that a new HDD will arrive next-day from Dell, and it's to go in server XYZ in rack V-U at drive position T. This may well be a free service, assuming normal failure rates.
Remote hands is a thing indeed. Servers also tend to be mostly pre-built now days by server retailers, even when buying more custom made ones like servermicro where you pick each component. There isn't that many parts to a generic server purchase. Its a chassi, motherboard, cpu, memory, and disks. PSU tend to be determined by the motherboard/chassi choice, same with disk backplanes/raid/ipmi/network/cables/ventilation/shrouds. The biggest work is in doing the correct purchase, not in the assembly. Once delivered you put on the rails, install any additional item not pre-built, put it in the rack and plug in the cables.