The process you're describing from your prior employer's IT department is generally only feasible for large organizations. This is primarily because they are able to exert a level of control over their computer systems which you cannot, as simply the "repair go to guy of the family", maintain.
Enterprise-level environments tend to be largely homogenous in terms of the hardware, software, and system configurations that are deployed. Usually, all of an organizations users are given only a handful of options from which to choose their computer hardware.
The operating systems and software installed on the computers is also largely standardized across the environment. Any deviations from the standard build are documented on a case-by-case basis, and the IT department either makes a special build image (if the software is widely, but not globally, used across the organization) or keeps the software installers ready at hand in their own repositories.
The IT department also usually keeps track of all the licensing, so they know which software keys are assigned to which computers/users. (Alternately, a Volume License may be used which will be the same on all systems.) Typically, any software which is not part of the standard build or otherwise documented by the IT department is beyond the IT department's responsibility to support - and in many cases, against company policy to have in the first place.
The configuration of the operating systems and software is standardized much in the same way as the rest. However, most of this is usually handled automatically by enterprise management systems (e.g.: Group Policies in Active Directory) instead of being manually done at build time.
User-level data is also generally not a big concern for the desktop support personnel. In most cases, users are advised (if not required) to keep all mission-essential data on company servers instead of their own workstations. So, when a rebuild is needed, the desktop support people shouldn't need to care too much about backing up the system first. If local user data is a concern, responsible organizations will have the end-user systems regularly sending backups to a centralized management system.
All of this starkly contrasts to the challenges faced by an at-home IT geek with when it's time to help out the family:
- You have little to no control over what hardware you'll be asked to support.
- You have even less control over what operating systems or software you'll be asked to support or re-install.
- You probably don't have all the license keys for all of the software you'll be asked to support or reinstall.
- Chances are, whomever you're supporting has lost some or all of the installation files and/or license keys for the operating system or software they need you to help with.
- There's no telling what special configurations you'll be asked to make on the operating system or software for any given user.
- Most of the people you'll support will have their data scattered across several folders on their hard drive.
- Most of the people you'll support probably aren't backing up their data to an external device.
It's not entirely hopeless, though. As some others have mentioned, there are solutions available to home users that can help alleviate some of these pains. However, be aware that there is not going to be a one-size-fits-all answer - in terms of addressing this question to a global audience, or even to a single "repair go to guy of the family". Chances are, you'll never get to a point where even a majority of your at-home system rebuilds take up anything less than "an entire afternoon". You'll have to adapt, with whatever tools are at your disposal, to every case as it comes to you.