I was going to link to a review of the literature on the local effects of new construction, which reviews the empirical literature showing that any additional demand is more than offset by additional supply, but I'm having trouble finding the PDF. (For global effects, remember that building new housing does not change the global demand for housing, as it does not create new people, so hopefully it's obvious that new housing supply must be a global downward pressure on prices).
But this newer paper is far more comprehensive on far more topics. IMHO it's far too charitable to some really bad and research with highly biased and methods (e.g. Damiano) but it still does point out the flaws there.
https://furmancenter.org/files/Supply_Skepticism_-_Final.pdf
Unfortunately this review does not cover the spate of papers in the 2020s that I was originally thinking of, so I'll see if I can dig that up still.