You might know this, but I wanted to clarify what is actually being done here. Nonlinear regression is usually solved computationally (and automatically, through software) by picking an initial model equation, plugging your data into the equation, and then comparing how far off your actual data is from the predicted result. It's an iterative process -- each time the computation is done, the model is updated with the results and the process is repeated. Once the data closely match the predicted result the final equation is plotted. This can then be used to extrapolate. It's NOT the same as plugging this into excel and choosing "line of best fit". Excel's best fit line is a very simplified computation!

The way you're describing it, it sounds like you are thinking of plotting with a polynomial model -- in this case, yes you can almost ALWAYS get a perfect curve, because polynomials are infinitely mutable -- you can always add more degrees to increase the accuracy. The same does NOT hold true for exponential decay! There is really only one line of best fit for any exponential plot, and that is what is found via nonlinear regression.

I hope that better explains what I'm trying to say. Your overall POINT is not wrong! They perhaps are a bit overconfident in their analysis, and it's very possible the data only APPEARS exponential now, and will fall out of that trend soon. That would go well with your reasoning. But I did want to explain that the process they used is legit, and a common analysis in scientific research.