Hrmm... So the assumption is based on the amount of C-14 in the atmosphere is the same found in plants/animals.
It's not an assumption. We know it's the same because of the carbon cycle - the carbon present in organisms is ultimately coming from, and going back into, the atmosphere. Photosynthetic organisms are fixing CO2 into sugars, which are consumed by other organisms and incorporated into their structure; organisms are respirating and breaking down sugars into CO2 for energy and exhaling it into the atmosphere.
That's the carbon cycle, in a nutshell, and it ensures that carbon is cycling through the biosphere so fast that the isotope ratios in living things will always be the same as the atmosphere. It only stops when the organism dies. And when it stops, the ratio of carbon-14 begins to drop due to radioactive decay. Therefore we can measure the time elapsed since an organism died, subject to the ability of our instruments to measure C14 and its decay products.