With all due respect, it does not look like you have looked at any of this earnestly. First of all, the results are not marginally significant. In every variant of the analysis, the results are significant at the p < 0.001 level. This is not modelling, this is data analysis. "Validated" doesn't really make sense in this context.
Methodologically this is bog standard econometrics. The quality of the dataset is as good as any used in economics, all you have done is cited their own transparent disclosure of the limitations. You can now say that we should never look at any economic data to make any analysis, but that would be a rather extreme position to take. Much much worse papers have had disproportionate policy impacts [1]. These papers are taken very seriously by every major relevant institution.
If the data is low quality, and you still obtain a strong signal, that shows the result is robust, unless the data errors can be expected to be correlated with the drivers. And this is where this analysis is a lot simpler and more robust than standard econometrics: The economy doesn't influence the weather. Common drivers are conceivable, but the panel regression controls for both regional and temporal background rates. At this point it is very difficult to conceive of common drivers that would not be controlled for here.
Finally: "One of the most depressing conclusions I reached when studying this topic a few years ago was that we don't really know if the world is getting warmer or not."
If your conclusion is contrary to that of _every_ scientist who studies this for a living, you should consider the hypothesis that the problem is with your reasoning/analysis and not with everyone else.
Methodologically this is bog standard econometrics. The quality of the dataset is as good as any used in economics, all you have done is cited their own transparent disclosure of the limitations. You can now say that we should never look at any economic data to make any analysis, but that would be a rather extreme position to take. Much much worse papers have had disproportionate policy impacts [1]. These papers are taken very seriously by every major relevant institution.
If the data is low quality, and you still obtain a strong signal, that shows the result is robust, unless the data errors can be expected to be correlated with the drivers. And this is where this analysis is a lot simpler and more robust than standard econometrics: The economy doesn't influence the weather. Common drivers are conceivable, but the panel regression controls for both regional and temporal background rates. At this point it is very difficult to conceive of common drivers that would not be controlled for here.
Finally: "One of the most depressing conclusions I reached when studying this topic a few years ago was that we don't really know if the world is getting warmer or not."
If your conclusion is contrary to that of _every_ scientist who studies this for a living, you should consider the hypothesis that the problem is with your reasoning/analysis and not with everyone else.
[1] https://en.wikipedia.org/wiki/Growth_in_a_Time_of_Debt