OS Ventura Installation Problems.

You should build a bootable USB installer from the Install macOS Monterey or Ventura installation image. Then boot from that flash drive and run disk utility First Aid on both the HDD and SSD that make up your Fusion Drive. If there is a problem with either of those disks it will cause a lot of problems. Performing First Aid by booting from a USB flash drive means the Fusion Drive won't be in use at all. Unlike using even Recovery Mode because Recovery boots from the Fusion Drive. You'll need a 16GB or larger USB flash drive. Hold the Option Key when powering on to boot from the Flash drive.

Create a bootable installer for macOS - Apple Support

Thanks, this identifies the root cause of your problems. But let's get more specific information.

Open up Disk Utility again, you don't need to be in Safe Mode nor booting from the USB drive.

Go to the View menu and select Show all Devices. Here's an example of what I see on my internal SSD on my sidebar:

When you expand out all the tree branches you can see there's the APFS 'Container disk3' which holds the 'Macintosh HD volumes' which holds a greyed out Macintosh HD which has a 'Macintosh HD snapshot'. Then you have the Data volume. The display of this information will likely vary as this is how Ventura displays it. For example, the Data volume might be listed as 'Macintosh HD Data'.

Regardless, what's happening here is the System volume is the read-only (why it's greyed out) 'Macintosh HD' and the snapshot is taken from that read-only immutable volume. The OS is booted from that snapshot which is signed and sealed by Apple. Meanwhile the Data volume is where all your data and other user data as well as applications you installed reside.

If the disk errors are on the System volumes then that might indicate a disk failure. Since you have a Fusion Drive in an iMac it consists of a small SSD volume paired with a HDD volume that is much larger. Any complete disk failures on either of those two drives can result in catastrophic data loss. I suspect your HDD drive has some bad sectors and it might be getting ready to die completely.

BACKUP ALL YOUR DATA ASAP to an external drive large enough to hold it all. You can use Time Machine or Carbon Copy Cloner to accomplish the task or you can just copy all your Desktop / Documents and anything else important off the Mac.

In order to confirm a hardware failure, a professional would need to wipe out the Mac completely, break the Fusion pairing, test both the SSD and the HDD. Identify and confirm there is a hardware disk problem. Analyze the S.M.A.R.T drive diagnostics, etc. Then if the HDD is dying as I suspect to be the case. I would highly recommend replacing the Fusion SSD / HDD with a single large SSD the biggest you can afford. You might not actually need 2TB's of internal storage. You might be able to offload data to external drives for archival, etc.

If you are lucky, there won't be any hardware failure and you can simply re-install Big Sur 11.7.2 from scratch and restore your data and upgrade to Ventura. But from experience, it's possible that 2TB HDD is getting ready to die and it's been experiencing problems for quite some time.

How many times did you run First Aid? If only once then try several more times. I've run into occasions wher3e it took 3=4 runs to fix reported errors.

If you already have run it multiple times then this post will self destruct in 10, 9, 8,………

When I say break the pairing of the SSD / HDD Fusion drives. There is a software level pairing of the two physical devices. That needs to be removed so you can see both individual drives instead of appearing as a single device. That way, Disk Utility can display the hardware S.M.A.R.T status on each device and you can run First Aid on each disk. Then you would erase both the SSD / HDD and recreate the Fusion pairing if there are no hardware failures indicated by errors in S.M.A.R.T. If there are hardware errors, then the physical disks need to be replaced.

Opening up an iMac is not an easy task even for a professional. It involves removing the front glass with special tools, breaking the seal and removing a glued gasket. Then removing the actual LCD panel behind the glass to access the system board. At that point you can replace the Fusion SSD / HDD disks with a single large SSD. Then put it back together and replacing the glued gasket properly and ensure you don't break anything. It's a daunting task. You also need to make sure you obtain the correct gasket and the proper glue, etc. I think Apple is making it easier with their new repairability changes to obtain the parts you need. Where in the past you were buying from 3rd party vendors and you didn't always receive quality. For example, there were reports of iMacs front glass falling out because the gaskets were not correct. Plus when you remove a gasket you need to scrape off all the glue and clean the surfaces with alcohol, etc. Again, attention to detail and doing things the right way is paramount to a successful repair.

I would highly recommend you work with Apple directly or an authorized Apple hardware support center. To get the Fusion drives repaired.

I am leaning towards potential HDD issues in the Fusion pairing and that's only going to get worse over time. It's very important that you backup your data, at least the critical data. Typically HDD's will continue to fail and eventually stop working altogether. If that happens you lose all your data in a catastrophic failure.

If you still run into the problem you could call them back. If it were me. I would go so far as to undo the Fusion so Disk Utility can see both the SSD and HDD then you can check the S.M.A.R.T status take a photo with your iPhone and explain to Apple that you have hardware faults on the Fusion SSD or HDD as proof. Of course that results in complete data loss so again, BACKUP is Step ONE!

You can identify the fusion pairing by using Terminal:

diskutil apfs list

Look for the "virtual disk" that is the "Fusion Drive" and you can then do this:

diskutil apfs deletecontainer disk#

Where disk# is changed to be the virtual disk Fusion Drive.

Then when you do diskutil list you will see both the SSD and the HDD listed. Then you can open the graphical Disk Utility and check both with First Aid and look for the S.M.A.R.T status to see if it shows any errors. If they both state Verified then they are working fine without a hardware fault.

Then you can follow the diskutil resetFusion instructions for Mojave or newer and that will re-pair the SSD and HDD back together as a Fusion Drive with APFS as the old Core Storage is no longer used.

Then you can install macOS clean and restore your data from backup.

Could still be some kind of soft disk corruption. But if S.M.A.R.T throws errors instead of 'Verified' then you've definitely got hardware failure on the drive(s).