I worked for an algae biotech company and while we did "dope" our ponds with CO2, providing ample mixing to get more sunlight and supplementing the ponds with plentiful nutrients (the "big 3" but also trace elements of a number of more obscure inputs) had dramatically larger impacts on growth rates.
AFAIK, the FACE studies do show faster growth with more CO2, but it's only observable at concentrations ~200ppm over ambient. [https://nph.onlinelibrary.wiley.com/doi/10.1111/j.1469-8137....]
In many parts of the world water is also pretty important. We know that global warming will change global air currents, but is pretty difficult to predict how this will affect precipitation.
Which is why you more often see stuff like deserts greening when they were dead before. The Amazon rainforest isn't going to 2x in height.
NASA Earth Observatory:
https://earthobservatory.nasa.gov/images/146296/global-green...