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.