Trouble upgrading to Catalina on late Spring 2013 27" Mac

Download and run Etrecheck. Etrecheck is a diagnostic tool that was developed by one of the most respected users here in the ASC and recommended by Apple Support  to provide a snapshot of the system and help identify the more obvious culprits that can adversely affect a Mac's performance.

Copy the report

and use the Add Text button to include the report in your reply. How to use the Add Text Feature When Posting Large Amounts of Text, i.e. an Etrecheck Report

IMPORTANT:

Before running Etrecheck assign Full Disk Access to Etrecheck in the app's Privacy preference pane so that it can get additional information from the Console and log files for the report:

ALSO IMPORTANT:

Then we can examine the report and see if we can determine the root of the problem.

Well that is the root of the problem. Mac OS does not benefit from nor does it require ANY third party Antivirus. The best security your Mac can have is Mac OS itself. All that AV apps to is to make Mac OS slow, unstable and appear to have bugs like what you are experiencing now. Please locate the developers uninstall instruction and uninstall the apps. Once uninstalled, restart in Safe Mode per the directions in Use safe mode to isolate issues with your Mac - Apple Support. Then restart normally and then re-test, I suspect the problem will be then solved.

I'm not sure what is going on, the AV apps you had installed are gone. However it's possible they damaged Mac OS when they were there.

What I think should be done is to install Mac OS over the version currently installed fill in any damaged or missing pieces. This is pretty simply to do and doesn't compromise your apps or data. However that being said, I don't see any evidence that this computer is being backed up. So what I recommend before doing the re-install is to buy yourself a good quality External Hard Disk like a OWC Mercury Elite Pro ( https://eshop.macsales.com/shop/owc-mercury-elite-pro ) and let Time Machine back up your computer. Once you have a backup in place then you can do the Mac OS install, the backup is something that EVERYONE should do as not having a minimum of one backup strategy is a guarantee you will lose data.

After you have a solid backup, then restart in the Recovery Partition (Command + R on startup) and install Mac OS. The backup at that point is your insurance plan.

What happens when you click on the Get Button in the App Store?

Also what is the model ID of your iMac:

Look in System Preferences.

Do you have any antivirus, cleaning or other third party "maintenance" apps installed?

We need to see the full EtreCheck report in order to evaluate it, there is no confidential information in the report.