Late 2013 27" iMac with new SSD not upgradable past Mavericks?

Find another Mac tech. There's no reason to believe it cannot be upgraded.

Your iMac 27" Late-2013 is very capable of running macOS 10.15.7 Catalina, but nothing newer.

How to Upgrade to  macOS 10.15 Catalina:

❗️– IMPORTANT  – Backup your data. (You’ve been warned.)

Learn How to Back up your Mac with Time Machine:

Back up your Mac with Time Machine - Apple Support

❗️ – IMPORTANT  Your 32-bit apps won’t work in Catalina.

Get this free utility to check the compatibility of your apps before you upgrade:

St. Claire Software – Go64: https://www.stclairsoft.com/Go64/

✅ - Download the macOS 10.15 Catalina installer here:

https://itunes.apple.com/us/app/macos-catalina/id1466841314

The download will place the "Install macOS Catalina" app in the Applications folder. Double-click the installer to begin the upgrade process. 

A Late-2013 iMac cannot upgrade past macOS Catalina 10.15.7. Confirm that it is a Late-2013 iMac via the Apple menu () > About this Mac, and what is shown inside of the parenthesis on the hardware description line of that panel.

Apple's macOS Catalina does not support older 32-bit application software, and you can determine your current applications that would be affected by a Catalina upgrade beforehand. Get the free Go64 application and run it to see the list of 32-bit applications and dependencies.

Next, perform a final Time Machine backup and then set that drive aside for safe keeping. Do not reuse it for the first full backup of Catalina, and instead, start with a new external SSD that is 2.5 - 3x the size of your current internal drive.

You can get the full macOS Catalina installer from How to download macOS - Apple Support. That installer will end up in your /Applications folder, where you can initiate the upgrade from Mavericks.

You should have your iMac serviced only by an Apple Store, or Apple Authorized Service Provider. Due to the age of your iMac, neither may service it now, and I would ask either for the name of a qualified Mac knowledgeable technician that they would recommend for future work.

If it was running Monterey, it was running a patched version which violates Apple's Software License Agreement and we cannot discuss or provide guidance about that here in the Communities. You'll have to look elsewhere for answers to that.

And that's why your tech couldn't upgrade it per your request.