Does your model without log(GNI per capita) basically just include a proxy for log(GNI per capita), by including other predictor variables that, in combination, are highly predictive of log(GNI per capita)?
With a pool of 1058 potential predictor variables, many of which have some relationship to economic development or material standards of living, it wouldn’t be surprising if you could build a model to predict log(GNI per capita) with a very good fit. If that is possible with this pool of variables, and if log(GNI per capita) is linearly predictive of life satisfaction, then if you build a model predicting life satisfaction which can’t include log(GNI per capita), it can instead account for that variance by including the variables that predict log(GNI per capita).
And if you transform log(GNI per capita) into a form whose relationship with life satisfaction is sufficiently non-linear, and build a model which can only account for the linear portion of the relationship between that transformed variable and life satisfaction, then within that linear model those proxy variables might do a much better job than transformed log(GNI per capita) of accounting for the variance in life satisfaction.
I’ve raised related points here, and also here with followup, about how exponential decay with a fixed decay rate is not a good model to use for estimating long-term survival probability.