Nobody knows. Which is why it's colossally reckless to propose anything around this as a guaranteed solution, instead of a proposal that needs more research.
This is irrational pearl clutching. If sulfur is added, it will be done gradually, with the results monitored. There would not be some sort of catastrophic cooling.
The real sticking point would be that some countries would not want the cooling. Russia, say.
So let me get this straight: in the face of a failure to control global warming by other means, you'd prefer your anxiety about possible risks to prevent any attempt to counter the warming by this approach, dooming the world to potentially catastrophic temperature increases?
Not really? We have no idea the relativity of our change vs what's supposed to happen when coming out of an ice age because of our short window we are observing.
We may change the climate in our entire history as much as a super volcano does, or many times less.
We have natural experiments where sulfur aerosols were injected. The 1991 eruption of Mt. Pinatubo, for example. It injected 17 megatonnes of sulfur dioxide into the stratosphere, causing a global cooling of 0.4 C and a northern hemisphere cooling of 0.5-0.6 C. This also affected ozone, which suggests aerosols other than sulfuric acid droplets might be a better idea.