"Last time I checked the facts, about 1/3 of all atmospheric carbon is removed each year"

This is kind of silly as a stand-alone statement. If this were the truth and the whole truth, atmospheric co2 concentrations should be falling, not rising steadily (as they clearly are).

Are we perhaps talking about residence time or turnover time?

A co2 molecule does have an average residence time in the atmosphere of about 5 years, but when it leaves, it typically swaps places with one in the geosphere or hydrosphere (e.g. if a co2 molecule is sequestered in a tree, another tree dies and decays, releasing co2). This turnover has been pretty stable over geologic time (it has clearly changed, but the change has been very very slow).

Massive burning of fossil fuels quickly puts lots of co2 molecules in the atmosphere, that may only stay there for about 5 years, but their return and sequestration (in trees or carbonate sediments or whatever) is slower than the rate at which additional co2 molecules are pumped into the atmosphere. So the residence time is ultimately irrelevant.