Would a volcanic winter trigger an oceanic "spring"?
I can't answer the question in detail but maybe take some of the tension, addressing the so called bottleneck hypothesis.

It was never really supported by archaeology, the mentioned article just adds to the evidence that human variety didn't suffer that greatly from the Toba eruption as the bottleneck hypothesis might suggest and that there was no decade long volcanic winter.

Example from lake Malawi after the eruption:
https://www.sciencedirect.com/science/article/abs/pii/S0047248417302750?via%3Dihub

Southe Africa:
https://www.nature.com/articles/nature25967

India:
https://science.sciencemag.org/content/317/5834/114

And to my knowledge no extinction signal in Neandertal population (they were few anyway) during oxygen isotope stage 4 in Europe.

Some think that it is time to put the bottleneck hypothesis ad acta. 

There is evidence that eruptions "fuel" phytoplankton, but not necessarily connected to volcanic winters.