MacOS Monterrey installation failed, lost 10gb of space

As I understand your question, the upgrade installation began but it failed for some reason. Now you have insufficient disk space to begin the installation. Is that correct?

In addition to Free up storage space on your Mac - Apple Support, Optimize storage space on your Mac - Apple Support also offers some suggestions.

If you are still unable to create sufficient space for the upgrade even after exhausting all those steps, the workaround is to create a Time Machine backup, completely erase the Mac, install the latest version of macOS (read • below), and finally transfer the contents from that Time Machine backup when asked. To learn how to use Time Machine please read Back up your files with Time Machine on Mac - Apple Support.

• To install the latest macOS that is compatible with your Mac, refer to Other macOS installation options in How to reinstall macOS - Apple Support.

2 cents here

Rebuild the Spotlight index on your Mac

What is “Other” storage on a Mac, and how can I clean it out?

OmniDiskSweeper Safe to use

GrandPerspective 

How to delete Time Machine snapshots on your Mac

Locate backups of your iPhone, iPad, and iPod touch

Notation - If the user is using a cloning software like Carbon Copy Cloner - suggest tweaking the Safety Net Feature in this software. It may be making addition Snap Shots that are not being Cloned to the Eternal Drive. If this should be the case, these Snap Shot could be using additional space on the drive 

The final word from Apple on Managing the " Other/ System Data “ Category

Other / System Data: Contains files that don’t fall into the categories listed here. This category primarily includes files and data used by the system, such as log files, caches, VM files, and other runtime system resources. Also included are temporary files, fonts, app support files, and plug-ins. You can't manage the contents of this category. The contents are managed by macOS, and the category varies in size depending on the current state of your Mac.