Sulfate aerosols already reduce the overall warming from raised GHG's, however the effect is more closely proportional to the rate fossil fuel burning that makes them, a process that also makes CO2; ie when that burning stops the cooling effect fades away quickly in days to weeks. But CO2's climate impact is more closely proportional to the accumulation of emissions over time; when the burning stops the warming effect continues and declines only slowly, over decades to centuries.
A hypothetical case of starting a whole lot of high sulfur coal plants all at once would expect to see an immediate cooling effect at full strength within days to weeks, with CO2 initially adding zero warming but increasing over time; at some point the warming will be equal to the aerosol cooling, taking temperatures back to where they started and after that, exceeding it. Stopping the coal burning would cause a rapid spike in temperatures, yet continuing it for the sake of the aerosols continues to add yet more warming. There is no fixing the warming by adding more coal burning, but it can give the appearance of reducing it for a time. It isn't quite a Zeno's race paradox, but has similarities; you don't get to a finish line, ever. And it doesn't address the effects of raised CO2 on ocean pH.
Fossil fuel aerosols remain predominately in the lower atmosphere and don't stay long.
Volcanic aerosols reach the stratosphere and the aerosols that do stay there longer - up to a few years - but would need to keep being added to compensate for the CO2 accumulation; it would take dedicated efforts that, unlike coal burning, makes no other income, that must be funded by levies or taxes that will be, as they always are, resisted and evaded.
In parallel with phasing out fossil fuel emissions doing so can reduce regional and global temperatures, by reducing sunlight intensity. Which can have other (unintended) consequences, including to agricultural yields, which may not be justly distributed.
As an alternative to phasing out fossil fuel burning it must be done at rates that not only exceed the current warming from accumulated GHG's, but must exceed that from further GHG accumulations.
Stratospheric sulfate aerosol dispersal remains a hypothetical possibility but it lacks support, including from a large bloc that sees the growth of low emissions alternatives as the most effective and cost effective response to global warming, that opposes actions that provide justifications for continuing to use fossil fuels.
It is unlikely to gain sufficient support and funding to make any difference to the climate problem within the time frames we are facing.