If anything, we're seeing the opposite results, where economists publish influential papers demographics, crime and social structure.
"When dealing with humans, linear regression is going to be good enough" is a huge assumption to make.
And if you're going to claim that economists are publishing influential papers in other fields - and especially if you're claiming that they're doing so in an unprecedented way, with no inter-disciplinary collaboration - please provide some examples. And if you're thinking of Freakonomics, know that no researcher takes Freakonomics seriously, and neither should you.
As for sociologists "invading" economics, they sort of are. Economics and sociology have quite a bit of overlap, and researchers from the two fields often collaborate. And any group researching economic phenomena, even an inter-disciplinary one involving sociologists, would be identified as economists, not sociologists, by people reading their work. Although David Graeber, an anthropologist, did write an excellent book on economic phenomena in "Debt: the first 5000 years", and it has done quite well. You could say that it's "influential".
Unfortunately, neoclassical economics also has wide political support among the people it benefits: wealthy people and institutions, e.g. banks. Which also means they get bankrolled (hah) much more than other social scientists, which means they get preferential treatment. E.g., this very "Nobel prize" in economy that this theead is about is funded by a bank.
The force to change economics qould have to come from within economics, perhaps from behavioural economics, or new Keynesian economics (the first one seems more promising), or even from movements like degrowth or circular economics. You can't expect a sociologist to fix a different field, and that wasn't the point. The point was simply that sociology doesn't suffer this embarassment because they are not burdened by ideological pressure backed by monies interest.
Fun fact: The neoclassical economic school managed to remove the word "political" from "political economy" at the turn of the 20th century.
But since you asked, here's an accessible overview: https://francescosyloslabini.info/2016/06/01/neoclassical-ec...
https://replicationnetwork.com/2021/01/06/reed-the-state-of-... -- "indeed, there has been an increase in the number of replications over time. "
https://www.banque-france.fr/en/publications-and-statistics/... -- "First, we find a moderate replication success, in spite of a data availability policy."
But still, that is interesting. Thank you for taking the time. Although you first source is a survey of expectations by authors on replicability over time, not an actual measurement of replicability.
Your second reference says there are historically more replication studies being published now than before (people trying to replicate other studies), not that more studies replicate.
As you might be aware of, the number of scientific publications in general is growing, so I would also expect the subset if replication studies to grow. This is not very surprising or meaningful, but interesting.
The third one is actually about reproducibility, the authors have taken open data sets published together with papers and see if they can produce the figures and final values in the papers from the original data sets. That is an entirely different thing.