FYI, trying to access files using the macOS installer is not easy. The others are correct, Target Disk Mode is the best option to access the data assuming the broken Mac's internal Hard Drive is healthy. If the internal Hard Drive is failing, then any attempts you make to access the data will likely make the drive failure worse where even an expensive professional data recovery service may be unable to recover any data. Be careful since you may only have one chance at recovering data from a failing drive.
Have you tried to run the Apple Diagnostics to see if any hardware issues are detected? Unfortunately the diagnostics rarely detect issues even when a hardware issue is confirmed through other methods. My first guess would be a failing Hard Drive. A passing diagnostic is meaningless here. There may be other ways of checking the health of the internal drive(s) depending on the model of the broken Mac.
I don't understand the fascination with Ventura here. Any version of macOS 10.13+ compatible with this broken Mac will work to access the data, but I would not be using the macOS installer to access the data unless there were no other options since that is a lot of work. And you overlooked using Internet Recovery Mode (Command + Option + R) to access the online macOS installer.
And you should start a backup regimen for your other computers and any external media (including the cloud) which contains important & unique data. There are a lot more new ways to permanently lose access to the data stored on the internal SSD of the recent Macs due to all of the hardware, software, and security changes. Chance of data recovery is nearly impossible in most cases.
