Problem reinstalling el Capitan on an old iMac after already erasing the drive

You are welcome

" recovery mode " from the internal Drive is not the same as Internet Recovery Mode

To perform this action will require booting from a Bootable Installer. 

The Bootable Installer can Only be performed on an Apple Computer 

This will have to be performed from a Qualifying Computer to run the version of macOS to be made on the Bootable Installer. Example : Bootable Installer of Big Sur would have to be done on a computer that Qualifies to run Big Sur.

Notation: If the computer being used to perform this action is Too New or Too Old to qualify to run the version of macOS - this computer can’t be used.

Alternative is, to gain access to a Qualifying Apple Computer from a family member, friend or associate.

Let me revisit this " Here is where I get stuck because all of my other Macs are modern ones and they won't allow me to run the InstallOSX.dmg file. " from a different angle

Using a M4 Mac Mini with Sequoia 15.5

I can download, from Apple, the .dmg file which when 100% downloaded is 6.2 GB

I can open the .dmg file as presented in 2nd image

Where upon opening the InstallMacOS.pkg present the 3rd image

That is where I stop and did not allow it to continue

It where you are getting stuck ?

I would think so, for obvious reason

2nd image

3rd image

Which Mac is this?

If it is so old that it cannot run any version of macOS later than El Capitan, a charity might not want it. El Capitan is so old that even the version of Firefox ESR that runs on it has not gotten “critical security updates” in over three years.

Try booting into Recovery Mode using Command + Option + R which should bypass the need to authenticate with your AppleID to confirm a previous purchase of El Capitan.

Do you have access to another working Mac from 2007 to 2015? If so, that Mac can be used to make a bootable macOS El Capitan USB installer using the instructions in the following Apple article:

Create a bootable installer for macOS - Apple Support

Also, here is an Apple article with links to the official macOS installers along with instructions for extracting the older installers into the Applications folder:

How to download and install macOS - Apple Support

Unfortunately all of these things must be performed on a Mac qualified to run macOS 10.11 El Capitan which is generally a model 2007 to 2015, but you can confirm the exact hardware compatible by using the following article:

https://eshop.macsales.com/guides/Mac_OS_X_Compatibility

Depending on the actual age of the machine, so older computer can boot to Internet Recovery Mode

Boot the machine using the 3 keyboard combo Opt + ⌘ + r

This will require an active and reliable internet connection to reach out to the Apple Servers

If possible, use an ethernet cable from computer to Access Point / Router

The Option + Command + r combo should offer the Latest Version of OSX / macOS this computer qualifies to run

@Servant of Cats makes a valid point regarding the usability of El Capitan.

There is another option to installing macOS. Installing Linux Mint onto the computer is a good way to re-purpose older computers since Linux works well with older systems which have more limited resources. However, most people are not familiar with Linux so that may be a problem. Linux Mint will allow the use of the current versions of many popular web browsers (Firefox, Vivaldi, Brave, Chrome) and also includes the LibreOffice suite by default (works a lot like MS Office).

You can easily test drive Linux Mint by selecting the "Live" mode option when booting the Linux Mint installer. Performance will be terrible this way, but it can give you an idea what the OS will be like. Performance will be much better when actually installed to the internal drive assuming the internal Hard Drive is still healthy (I doubt it is still healthy).

Before installing Linux Mint to the internal drive, I would definitely try getting El Capitan installed so that you can create a bootable macOS 10.11 USB installer so you can easily go back to macOS if Linux Mint is not a good fit.

Did you ever make a bootable clone backup of your Mac, using Carbon Copy Cloner or SuperDuper!?

If so, do you still have the external backup? Such a backup could be pressed into service as a startup drive, and you could work from it while installing a clean copy of macOS on the internal drive.

Welcome

Alternative is, to gain access to a Qualifying Apple Computer from a family member, friend / associate, Library, Internet Cafe