Given that the results seem sensitive to arbitrary changes to how you analyse the data, might be worth doing some sort of multiverse analysis : https://www.martinschweinsberg.com/multiverse-analysis-exploring-the-entire-universe-of-analyses/
Second, I feel like there are 2 separate questions being discussed here.
Many of these papers ask the question "have the historic increases in GDP seen throughout the world been associated with increases in happiness?"
What we're actually interested in is "does increasing GDP increase happiness? And under what circumstances?" These are questions of causal inference, and so regression analyses with few to no control variables can be really misleading.
Imagine the following scenario: GDP has increased, which makes people happier, but the way we have achieved higher GDP has been at the cost of rising income inequality, declining leisure time and greater environmental damage. All of these have largely offset the increase in happiness that GDP would otherwise bring. In this scenario, GDP can, * in theory* increase happiness, but only if it's achieved without compromising on other things.