Virtual memory in user space (Operting Systems)

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Wouldn't a similar approach for a disk driver be possible?

I don't think so.

In most cases, the boot loader starts executing the kernel in supervisor mode.[8]

In computer terms supervisor mode is a hardware-mediated flag which can be changed by code running in system-level software. System-level tasks or threads will have this flag set while they are running, whereas user-space applications will not. This flag determines whether it would be possible to execute machine code operations such as modifying registers for various descriptor tables, or performing operations such as disabling interrupts. The idea of having two different modes to operate in comes from “with more control comes more responsibility” — a program in supervisor mode is trusted never to fail, since a failure may cause the whole computer system to crash.

Supervisor mode is “An execution mode on some processors which enables execution of all instructions, including privileged instructions. It may also give access to a different address space, to memory management hardware and to other peripherals. This is the mode in which the operating system usually runs.”

Like I said before, a hard disk doesn't have to have an OS on it, but if it does it ought to run its driver in kernel space, otherwise the very device that contains the OS is not part of the kernel and that makes no sense. Learning how your computer boots may help you answer some of your own questions.