I think geo engineering is not heartily pursued out of an abundance of caution: having established we can seriously damage the climate, many of the solutions that try to put the breaks on look even riskier because we don't want to overshoot. Or rely on unproven technology when the crisis is here, now.
(Not to be flippant, but the relevant xkcd: https://xkcd.com/349/ describes this well: sometimes things get a lot worse when an engineer tinkers with them.)
Geoengineering by taking excess CO2 out of the atmosphere, using machines that we can simply switch off, and storing the CO2 in deep basalt formations where it turns into rock, is almost certainly lower risk than leaving the excess CO2 in the air.
That doesn't mean we should rely on this entirely and not attempt to reduce emissions. But we're also past the point where we can rely on emission reductions alone.
If we don't have the will (budget) to stop emissions at source, why would be have it to go out and sequester it from the general atmosphere? Isn't that harder technically, more expensive and more energy consuming etc?
Distributed costs and concentrated benefits. Each particular emissions reduction plan impacts specific groups that will fight it. General taxation to pay for geo-engineering projects impacts everyone as well, but not in specific ways.
There might be more public support in a moon-shot big engineering project, and fighting the small-government types, than say, reducing emissions for factories and fighting the relevant lobby.
That depends. Cars are on the verge of being pretty cheap to electrify. Long-haul airliners, not so much. The smart thing would be to put a substantial price on carbon, give credit for sequestering, and let the market do the rest.
In the long term, we're going to need net negative emissions anyway. CO2 is already too high and is still rapidly increasing.
(Not to be flippant, but the relevant xkcd: https://xkcd.com/349/ describes this well: sometimes things get a lot worse when an engineer tinkers with them.)