I'd argue for elephants as a semi-domesticate. They were consistently put to significant economic and military use through the independent invention of at least four societies - Mesopotamia, Punic North Africa, North China and India, with Southeast Asia being a potentially fifth independent semi-domestication.
The general rule is that elephants were captured in the wild, enculturated to humans or with mahouts, and employed in large numbers by humans. Their use may well have been contemporary with or even preceded the domestication of draft animals.
The trouble with elephants is that they live too damned long, 65 years, making for a working lifespan of 20 to 40 years or more. That means that you don't really have a pressing need to continually invest in new elephants. The other problem is that they take so long to grow, 25 years or literally a human generation, that the investment will seem counter-productive.
There's evidence that all 'Elephant using' societies did breed elephants, and in particular, the most successful surviving one in Southeast Asia puts a fair premium and invests a lot of ceremonial and cultural energy into raising elephants.
But the reality is that for the actual economic needs, it was always cheaper to continually harvest wild elephants and semi-domesticate them or integrate them into human society than it was to breed them.
So the problem was that once the wild population gets hunted out or driven off by habitat destruction, then you need to make a significant social investment in breeding them. Unfortunately, their working life is so long that the investment in breeding doesn't make short term sense. Instead, the economics favour faster breeding, faster growing, though shorter lived creatures.
It's possible that Elephants in a historical sense carve out the economic niches and roles that are later filled by other domesticates.
As a result, working elephants quickly decline and vanish from the population within a few generations of the elimination of the wild populations. This happened in Mesopotamia, in north china and in north africa.
It might be theoretically possible to come up with a timeline pod, and circumstances where any or all of these cultures decided to make the investment in elephant breeding, in addition to or as an alternative to other draft animals. In which case, we'd be saying the Elephant is a domesticated animal.
It's worth noting that a number of elephant species co-existed with neolithic man - including presumably potentially faster living dwarf species. These include the mammoths (browsers), mastodons (grazers), the stegodonts of east asia, and the gomphotheres who lasted in South America right up until about 500 CE. Potentially, any or all of them may have been semi-domesticable in the same way that the African or Indian elephants were. But there was no need, in the cultures of the time, for their labour.
This is one of the factors that Diamond overlooks. Not only does the animal need to fulfill a set of requirements (and Diamond's specifics are in a number of particulars questionable), but the human culture has to actually have a viable role for them. And the animal has to be able to survive in the wild in proximity to the human culture, in sufficient numbers for domestication to be a possibility.
The Meso-Americans weren't particularly inept at domestication. The trouble was that all the potentially domesticable species in proximity got hunted out before domestication events could take place.