Edit history doesn't appear in back up file

An exported image (unless the special purpose file type 'Original' is chosen) includes picture information that reflects the edits you have done in LrC. Those edits are what needed to be done to some starting picture information, to bring it to this edited appearance.

Let's say this included only two steps: lightening the shadows X amount, and cloning out a bird in the sky. Your export includes lightened shadows, but does not also include the dark shadows as they looked before they were lightened. And it does not include any sky as that looked before your cloning, with a bird in it.

Hence you cannot now revisit those edits and redo them differently. Not from what's in this export. 

Edit information may tell you something has been cloned out from the upper part of the image (but was it a bird or a plane, perhaps Superman?) If there was edit history too, you would also know whether this mystery cloning happened before, or after, some shadows-lightening was done. But I can't imagine what anyone would do with that extra knowledge.

Edit history belongs within a workflow that preserves an unedited copy of the image (the starting picture information, still as first imported), and a Catalog that preserves all the subsequent work of its management and editing.

When I export an edited image file for back up onto an external drive, the edit history does not show up

By @Barry5E99

Exporting does not create a backup of the original, it creates a new file with the edits applied (baked into the file).

LrC never changes the originals, the edits are saved as text in the catalog, and are only applied to new files.

You need to back up the originals, and certainly not delete them.

If you need to save space, do not export, unless you need an exported file for a particular purpose.

Your suggested workflow of deleting the original after exporting, would be like throwing away a film negative after having a print made from it.

To get a better understanding of how Lightroom Classic works, read some of the articles here:

https://www.lightroomqueen.com

Here's an article I've witten:  Lightroom (Classic) basics

'History' only exists in the Catalog. It does not get included in an exported photo. Only the final appearance of all editing steps is 'burned into' an exported image.