“...than they have” or “...than they have done”?

Are you a British English speaker? There are differences between American and British English when it comes to ellipsis and do-support.

This page gives the examples:

(1) I ate all the chocolate, even though I shouldn't have done.
(2) I ate all the chocolate, even though I shouldn't have.

Sentences like (1) occur in British English, but would sound odd in American English, where (2) is preferred.

Now we can change the sentences to use a comparative construction (as in the question).

(1') I ate more chocolate than I should have done. [BrE]
(2') I ate more chocolate than I should have. [AmE]

Note that done in (1) and (1') is referring to the main verb of the first part of the sentence, ate. This is a special function of do known as do-support (also seen in sentences like "I ate chocolate and so did you").

An alternative to do-support is to repeat the first verb outright. This is fine even in American English:

(3') I ate more chocolate than I should have eaten.

Now going back to the sentence in the question:

people do more writing than they have at any other time in history

This has a structure like (2'), which is fine in American English. The main verb here is actually do, so it is also fine to say

people do more writing than they have done at any other time in history

even in American English. The structure is similar to (3'). Or, in British English, this could be interpreted as do-support, like in (1').

"To avoid repetition [of the main verb]

We use auxiliary verbs to avoid repeating the same verb or verb phrase in a sentence.

If the first part contains an auxiliary verb, we use the same verb in the second part.

I don’t like going to spas, but my wife does. I thought I had lost my keys, but I hadn’t. Tom can play the guitar and Tina can’t.
She has never had to face such difficulties, but I have. If the first part doesn’t contain an auxiliary verb, we use do or did in the second part.

She shares your opinion, but I don’t. Some wanted to go on with the plan, but others didn’t."

use of auxiliary verbs

It is correct as it is. But, it feels awkward because there is a "do" in the first part but no "do" in the second part. It's a "missing top step."

We live in an age when people do more writing than they have (missing do) at any other time in history.

People are do-ing in the first part of the comparison, but have-ing in the second part. It's awkward but correct.

I suspect the person who wrote that sentence is trying to emphasize a particular kind of writing. Thus they want to say that a person is to "do writing" rather than just write. Maybe they want to emphasize literature as opposed to jam-jar labels.

So the awkwardness has crept in under cover of this motivation. Don't have people "do writing." Just have them write.

We live in an age when people write more than they have at any other time in history.

There's no need to add the word 'done' here, it's just like the usual way of avoiding repeating word (here we mean 'verb'). The phrase 'at any other time in history' is a common one, people wouldn't mess it up with the former phrase 'they have'.

It is called ellipsis, basically when words are omitted from a sentence or language because they can be understood, or are implied by context or syntax. It's probably a good thing that it sounds a bit awkward to you because that means you can see that something is missing, and there is nothing wrong with including done in your sentence. Using ellipsis is not a requirement of English but it does help to avoid repetition, which is probably how it developed in the first place. Tᴚoɯɐuo is really onto it in his comments and he may have something to add the question of how it developed.

see here:https://en.wikipedia.org/wiki/Ellipsis_(linguistics)