Sequestration underground doesn't really work. It is based on lies. It will just bubble back up when the Earth's plates move. For capture to work, it has to be reacted with a stable absorber, so it cannot bubble back up into the atmosphere.
We have a worked example for geologically stable carbon sequestration without any novel chemical bonding, and that's storing carbon in compounds that are mostly unhydrogenated carbon by mass, deep underground:
Charcoal. If you aim to sequester carbon without some kind of reactant (and most reactants are incredibly energy intensive to make & stage, burning more CO2 than captured), you have to effectively make charcoal. Growing a forest, pyrolyzing it, and burying the charcoal, is the inverse process of coal mining, and is the default comparator on cost, effort, and materials for any sort of carbon sequestration scheme.
There is not nearly enough available space on the surface of Earth to make any dent into the atmospheric carbon dioxide concentration over any relevant time frame. We burnt through millions of years of sequestrated carbon in a century and some, but we do not have millions of years to do the sequestration again.
If we solve fusion, or even if we exploit space solar on a massive scale, we should in principle have all the energy we need to put the carbon back into charcoal.
Sure, with enough energy we can do whatever we want, but we just do not have enough clean energy, that is the very problem carbon capture is supposed to address.
It would be completely disingenuous to use carbon capture as an excuse to continue the use of fossil fuels. Firstly, it will be so price inefficient. Secondly, it delays the inevitable, meaning the transition to clean energy, as some European countries already have accomplished.
The demand for carbon capture makes more sense after we have already transitioned to clean energy, as we can then begin to reverse the damage that excess CO2 and methane have been causing. The financial model for doing this is by a long-acting insurance firm that not only collects insurance but also uses the premium to preemptively take restorative actions.
Carbon capture is of course less efficient than not burning fossil fuels to begin with, so the first goal must be to no longer emit new carbon. Then it may make sense to capture some carbon and turn it into synthetic fuels for certain applications because of the energy density and ease of storage. Maybe even for plastics and similar stuff. After that, undoing a century of emissions, that is a monumental challenge. Depending on how exactly we want to store the carbon, we might need more or less the same amount of energy that we got out of fossil fuels over the last century. Double our renewable energy generation capacity and then use half of it for half a century - roughly taking into account the increase in emissions over the past century - to capture carbon.
That's almost certainly a dumb idea. Always was. A red herring from the fossil fuel companies.
Obviously digging 1000kg of coal out of the ground, burning it to make electricity, and using that electricity to _synthetically create more coal_, and putting that back deep underground, is not going to put away even 100kg of carbon; You're probably lucky if you get 10kg given that CO2 is a trace gas.
You can do it cheaper, if you're willing to tolerate the carbon having less than the geological stability of coal. But then the ground burps and you're back to square one.
The part of carbon sequestration that isn't insultingly is the question of what happens once we've destroyed all the fossil fuel infrastructure and replaced that part of our civilization with renewables and (since it's always going to be advantageous to be a free rider) a good number of bomb craters.
My suspicion is that even when we get to that point, sequestration of 1kg is _so much more difficult_ than emission of 1kg was, that there's still a vast gulf between being at that point, and the point where we have the excess resources sufficient to sequester significant portions of the atmospheric carbon artificially. Probably instead, we end up coasting for a long while to see how much the biosphere can put away on its own; If we have climate change issues, we deal with them via stratospheric sulfur injection and by stochastically just adapting our footprint on the planet.
But I'm willing to be persuaded on this point. Perhaps there's work to be done with some kind of advantageous catalyst, mineral reserve & reaction (olivine maybe?), some way to cheat the system in the long run. I just don't think it's especially relevant to our plight in the short term.
The point is, charcoal production is the default method of carbon sequestration, the reliable null hypothesis by which you have to measure any other method, the denominator of human effort in this venture. No, even charcoaling the entire Earth's forests is nowhere close to sufficient to offset coal production (how the fuck could it be? The Carboniferous epoch was 60 million years long and we're going to run through its detritus in less than 600 years!), but a project to run a country-sized forest through the oven and into the mines every year is the _baseline_ for future meaningful effort.
CO2 levels over 600 ppm indoors already start to hurt cognitive function and cause drowsiness in me. Due to the cold, I have been managing with 550 ppm in the winter, otherwise with 500 ppm in the summer. A decade ago, the indoor level easily was 100 ppm lower. This is becoming a problem fast as I run up against the 600 ppm threshold. The older you get, the more intolerant you become to high CO2 if you want good cognitive function.
The point is that no, sulfur injections aren't going to fix climate change, nor will a solar shield. The reason is that they leave high CO2 on the ground which harms cognition. One can't really make a living with a deteriorated mental function.
Burning fossil fuels for energy to then unburn carbon dioxide is of course idiotic, one has to use renewable energy sources. If you could make more synthetic fuel than you put into the process, then you would have a perpetuum mobile. That is also why my comment says clean energy. But even then, read my other comments and you will see that I am extremely sceptical because of the scale of the problem.