According to Wikipedia, Earth atmosphere weights about 5e18 kg. A m^3 of air is alightly above 1,2 kg under normal conditions.
Suppose that on average each person from the most resourceful 1 billion people on Earth installs a device for air cleaning. The machine sucks air in, removes CO2 and releases the clean air. The device works using renewable energy, so by itself it doesn't contribute additional CO2 to the atmosphere. Suppose we want to remove 10% of CO2 from air, or in other words we want to pass 10% of Earth atmosphere through such cleaning devices. The year has about 365 * 24 * 3600 seconds, so to have those 10% in 10 years each device has to process 5e18 / 1e9 / 365 / 24 / 3600 / 10 / 10 = 1.6 m^3 of air per second.
This is not a small number, yet it's not immediately obvious why this isn't possible. For example, energy requirements can be small, because the device effectively doesn't change the bulk of the air. 10 years isn't that much comparing with the speed of raising air temperature. Cost of such a device could be perhaps smaller than a modest car. We still have option to deploy more devices, to separately clean ocean water etc. This is largely technical and not political solution.
What's wrong with this approach?