You could be right, but I think the more likely outcome is a 'Ministry for the Future' scenario where India or China just decide to unilaterally implement geoengineering and dare the rest of the world to stop them.
I don't know if any country has the industrial base to do it yourself, as you say, but if any does, it's almost certainly China just due to manufacturing output and expertise alone. They're, what, 30-35% of the world's output? Set aside 30% of that and they can still feed a lot of exports to other countries - more than anyone else - to bring in money. And nobody else has anything close to the leading-edge hands-on know-how to build all the stuff that would be required. As you mention, there's a lot nobody has yet. So the country best at building cutting-edge hardware is gonna have the head start.
If you're a western pessimist, that whole mega-project could easily unlock even more manufacturing, materials science, etc, progress that could be applied to the rest of their output too, furthering the gap between them and everyone else. (But I guess if they did single-handedly save however many billion people from displacement, and the majority of developed countries from massive crisis/invasion/etc, that's probably the least of our worries!)
It looks like we are in agreement about rough requirements, but estimate consequences differently. That's precisely what I was talking about - even for China powerhouse it would require 30% of GDP (example number, but it's good enough). My point was that 30% of GDP is an incredibly, astonishingly expensive amount of money to be diverged to a completely new activity.
Just to illustrate - NATO countries are currently haggling about spending even 3% on a military out of all things, and military spending is famously popular in most of the countries among people of power, due to all the corruption and jobs and factories and manufacturing and prestige etc. Poland ramping up spending to 4-5% is seen as extremely overspending (it's rational for survival reasons, but economy may not bear that long term). 30% spending on a thing not benefiting rich class in short to mid term is anathema. Not even grandpa Xi with his totalitarian power would manage that.
Also let's not forget that grandpa Xi has this tiny little victorious invasion of the independent Taiwan planned, likely done in a double push together with his puppet Kim against Korea, and simultaneously conducting a total blockade of the South China Sea for a many years. China may even manage that, but that won't be profitable in the slightest, and during the war all that economy engine will stall due to disrupted shipping and sanctions. They just won't have those 30% of GDP to spend on a side.
That's not outside of the capability of a country like India, much less China
Now that I think about - this is actually a motivation for China to do all that expenses. Establish themselves as a monopoly caretaker of the program, wait out until alternative programs will ramp down due to lowering interest, and then start blackmailing the world to conform to their demands or else they will stop injecting sulfur. That would be an incredible power play :) .
That's not nothing, but it's definitely something India could afford to do unilaterally, not to mention China