OS X Base System Locked

I don't think you can do an Internet Recovery on a 2010 iMac. I'm surprised it didn't pop an error at the start. I just checked and your model came with 10.6.3 Software Restore DVDs. That is what you will need. Do you have those disks? If you do, then put Disk 1 into the optical drive, restart the computer while holding down the "C" key to boot from the DVD.

1. Boot the computer using the Disk 1 that came with your computer.  Insert the disk into the optical drive and restart the computer. After the chime press and hold down the  "C" key.  Release the key when you see a small spinning gear appear below the dark gray Apple logo.
2. After the installer loads select your language and click on the Continue button. When the menu bar appears select Disk Utility from the Utilities' menu. After Disk Utility loads select the hard drive entry from the left side list (out-dented entry - type and size info.)  Click on the Partition button in the Disk Utility toolbar.  Set the number of partitions to one (1) from the Partitions drop down menu, click on Options button and select GUID, click on OK, then set the format type to MacOS Extended (Journaled), then click on the Apply button.
3. When the formatting has finished quit Disk Utility.  Proceed with the OS X installation and follow the directions included with the installer.
4. If you are planning to sell or give your computer away, then do the following: After you reformat your hard drive and reinstall OS X, the computer restarts to a Welcome screen and asks you to choose a country or region. If you want to leave the Mac in an out-of-box state, don't continue with the setup of the system. Instead, press Command-Q to shut down the Mac. When the new owner turns on the Mac, the Setup Assistant will guide them through the setup process.
5. If you plan to keep the computer, then you need to first install Snow Leopard. You will have to download and install the Mac OS X 10.6.8 Update Combo v1.1 to update Snow Leopard to 10.6.8 and give you access to the App Store. From there you can upgrade to the last supported version of macOS for your model.

Did you check and reformat the disk before installing macOS? It shouldn't take more than one or two hours over Wi-Fi; faster if using Ethernet?

Formatting doesn't write to the whole Disk, just zeroes out the Directory Sectors. :)

That is not your HDD or SSD. That represents the mounted disk image in the Installer. Choose the proper destination which will be above the line separator.

Edit: follow Kappy's direction.