The idea of a future American administration unilaterally deciding to tinker with geoengineering and somehow making things 100 times worse, is one of my nightmare scenarios.

Back of the napkin, thinking out loud, pumping a bunch of sulfur into the stratosphere might mitigate for a short-term the worst effects of climate change. Same with a few other compounds. Like OP replied, it would also mean that for the next several centuries we're going to have to continue to pump those compounds into the stratosphere, and it also assumes there won't be knock-on effects that we are simply ignorant of at this time.

The problem with geoengineering is we only have one biosphere. You can't really do experiments, you can't know for sure what affects whatever action you take will have around the globe. This, in addition to it being relatively cheap and easy, being the kind of short-term solution that American politics loves, leaves me certain that it will be an American president who finally darkens the skies.

https://youtu.be/AEagAcFP1II

Yes very likely.

I don't really know where people's thinking is at, as I don't follow this stuff very closely to be honest. I've just listened to a few talks about this kind of stuff. The double edged sword of the geo-engineering stuff is that it's relatively cheap. A couple billion at most, maybe if that? That covers basically every state and an uncomfortable amount of individuals, so basically any state feeling the pressure and not willing to wait around for richer countries to do something could pull the trigger on something like this. The biggest problem, imho, is that once someone has started it you can't really stop it because of the aforementioned termination shock. You'd have to keep pumping the upper atmosphere with aerosols until you bring GHG levels down, otherwise you go from your nicely engineered temperature to whatever the temperature would be without aerosols within a matter of months(I believe, don't quote me, but it's relatively fast) giving basically no adapation time for humanity, let alone the natural world. The other major problem is that if you engineer a lower temperature, the feeling of urgency may subside and lead to states becoming complacent in reducing GHG emissions. I think geo-engineering shouldn't be seen as a solution, but rather as a very last resort emergency brake kind of measure. However, given that the only state on the planet to seemingly give any fuck about any of this is China, I imagine we'll see that trigger pulled at some point in our lifetimes. Whether the global community will agree with doing it at the time will be a different matter, but it would be exceedingly difficult to stop or prevent anyone with the resources from going through with it.