This has been studied, and the research doesn't seem to support this view. Here's a good review of the recent research that finds that increasing housing supply slows rent growth and in some cases even causes rents to fall and that gentrification doesn't cause significant displacement of low-income households. [1]

[1] https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4629628 (You have to make a free account, annoyingly, but you don't have to pay anything to access the paper.)