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It’s not just that man-made emissions don’t control the climate, they don’t even control global CO2 levels. No graphs are available from Salby's speech or paper yet.CO2 variations do not correlate with man-made emissions. This graph comes from Tom Quirk's related work (see below).This means that we would need to know the natural upward and downward fluxes to much better than 3% to say that humans are responsible for the current upward trend in atmospheric CO2.
So if the C13 to C12 ratio is falling (as more C12 rich carbon is put into the air by burning fossil fuels) then we can’t know if it’s due to man-made CO2 or natural CO2 from plants.
Some years from January to January there may be a rise of 0 ppmv (ie no change), some years up to 3 ppmv.
If those changes were due to man-made CO2 then we should see more of those rapid increases in recent times as man-made emissions increased faster.
Many scientists have assumed that the net flows of carbon to and from natural sinks and sources of CO2 cancel each other out, but there is no real data to confirm this and it’s just a convenient assumption. Smith covered this point well in 2009 “It is often asserted that we can measure the human contribution of CO2 to the air by looking at the ratio of C12 to C13.
The problem is that even small fractional changes in natural emissions or sequestrations swamp the human emissions. The theory is that plants absorb more C12 than C13 (by about 2%, not a big signature), so we can look at the air and know which came from plants and which came from volcanos and which came from fossil fuels, via us.
In other words, temperature controls CO2 levels on a yearly time-scale, and according to Salby, man-made emissions have little effect.