Yes, CO2 is what plants eat, but it doesn't tend to be the limiting nutrient very often on Earth today. For the ocean (where 50-85% of photosynthetic oxygen production, and equivalently carbon burial, happens [1]) there are actually maps of this [2], and you may note that CO2 does not appear. On land, it's usually water, nitrogen, or phosphorus [e.g., 3].
Accordingly, there have been many studies on CO2 fertilization, but since it's not the limiting nutrient it doesn't tend to have much of an effect [e.g. 4]. At best there are benefits for some plant species in areas where water is the limiting nutrient, since higher CO2 means plants can keep their stomata closed more [e.g. 5], but even this is a bit of a mixed bag since dry places tend to get hotter and drier under global warming.
[1]https://en.wikipedia.org/wiki/Phytoplankton#Oxygen_productio... [2] https://media.springernature.com/m685/nature-static/assets/v... [3] https://nph.onlinelibrary.wiley.com/doi/full/10.1111/j.1469-... [4] http://www.pnas.org/content/early/2016/12/07/1610156113.shor... [5] https://nph.onlinelibrary.wiley.com/doi/full/10.1111/nph.124...