You may want to post this on r/AskAnthropology as this isn't strictly a history question (history is concerned with the written past and domestication predates writing). With that said, there are still some of us here who can answer your question, at least in part.

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First, just because an animal was domesticated does not mean it wasn't also food source. Every domesticated animal, dogs, cattle, sheep, goats, horses, etc. have been raised in part for food. Certain areas of the world still consume dog meat today. However, my particular interest has always been horses so I will focus on them.

Horses are believed to have been domesticated on the Eurasian steppes around 3500 BCE, though this date is debated and may have happened independently across a range of time and space. Before domestication, the horse was a major part of human diets on the steppes. This was because horses were extremely prevalent on the steppes. Much like native tribes on the plains of North America hunted buffalo because of how plentiful they were, tribes on the steppes of Eurasia hunted horses. Equid populations did exist elsewhere in Europe and Asia, but their populations were limited. Only the Eurasian steppes had them in numbers and it is here that they featured heavily in the bones left behind in garbage dumps by the ancient peoples.

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Once domesticated, horses were still a staple in human diets. One such site of early domestication is Botai in Kazakhstan where more than 90% of the animal bones recovered from the site were from horses. It is unknown when riding became commonplace, but their foremost use among the steppes peoples, like at Botai, were food and leather. While other animal populations like sheep or aurochs (ancestor to cattle) were also found on the steppes, horses were preferred because they were somewhat better suited to the winter months. Where cows and sheep will rub their noses raw brushing away snow and ice, horses use their hooves. This is likely why humans were interested in 'domesticating' or at least managing them in the first place.

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Now, why we didn't consider other species for work? Well, anthropologist David Anthony theorizes that humans only domesticate animals which they are familiar with, which is part of why domesticating horses happened on the steppes in the first place. Predatory animals like lions or bears would have been significantly harder for early humans to work with because early humans spent most of their effort removing or avoiding them. Lion hunts, like the one depicted in the Palace of Ashurnasipal II, used to be rather commonplace throughout the ancient world. Now, the Asiatic lion is limited to only a few locations worldwide. Compared to most domesticated animals, predators also never reached the same population levels to facilitate contact (and bears tend to be solitary anyway).