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.
It baffles me that my career trajectory somehow managed to insulate me from ever having to deal with the cloud, while such esoteric skills as swapping a hot swap disk or racking and cabling a new blade chassis are apparently on the order of finding a COBOL developer now. Really?
I can promise you that large financial institutions still have datacenters. Many, many, many datacenters!
Software development isn't a typical SME however. Mike's Fish and Chips will not buy a server and that's fine.