> the mean difference

Yes, but later the article shows that "mean difference" is really mean difference divided by variance.

> an important point is that whether something qualifies is really a question of the scale

It's one of the concerns, but I would say it's the main one. An effect size needs to distinguish between the two cases here, both having the same expected difference: https://imgur.com/a/hg2NNl2

If you care about units, you can talk about the expected value/difference, but that doesn't make that a meaningful effect size. What you need to do in those cases is to, at least, mention both the expected difference and the variance.