IMHO, you’re all quite close to the truth. Your polynomial is the Taylor expansion of Exp[(t-t0)/tau], as known since 1985 [1]. Following the current knowledge, the exponential increase of xCO2 is a “positive” feedback in the earth system. Initially, the perturbation was the anthropogenic CO2 emission while nowadays it is a mixture of emissions, other destructive anthropogenic actions and system inherent feedbacks. The system may have already left the basin of attraction of the Holocene. It may move towards another attractor, the hothouse earth [2] or the runaway greenhouse ([3] “ticket to Venus”) state.

1. Bacastow, Keeling (sen.) and Whorf (1985): https://doi.org/10.1029/JD090iD06p10529
2. Steffen et.al. (2018): https://doi.org/10.1073/pnas.1810141115
3. Kasting (1988): https://doi.org/10.1016/0019-1035(88)90116-9

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Just for the sake of correctness:

The used Keeling data are to be understood as results of measurements done in the middle of the respective month - so, for working with the precise data one has to replace the second line of my code by:

dataKeeling = TimeSeries[{DateObject[{#1, #2, 15}], #3} & @@@ dataRaw[[62 ;; -9, {1, 2, 9}]]];

This means a shift in the data of 15 days. Sorry for this flaw!