Not surprisingly, this observation isn't new, and there is much published research. It does not support your conclusion that there is a simple relationship. Quoting from
https://www.sciencedirect.com/science/article/pii/S026427511... which looks at that connection specifically within the NJ housing market (emphasis mine):
> This study replicates, using 2010 Census data, and extends past work on moderately rent-controlled municipalities in New Jersey, which began policies to stabilize rents while allowing landlords modest returns in the 1970s. .... We find that these 40-year-old policies do not exert any statistically-significant effects on their communities’ housing markets once other factors are controlled—a finding which has implications for affordable housing and advocacy in New Jersey and beyond. ...
> Rent control has been a hotly debated regulation in the U.S. for nearly 75 years (Fogelson, 2013; Gilderbloom, 1981; Gilderbloom & Appelbaum, 1988; Glaeser & Luttmer, 2003). This literature review concisely reviews the evolution of rent control, findings from previous studies of rent control’s effects, and the nature of rent control ordinances in New Jersey. ...
> When one controls for other factors affecting rents and rental unit quality and quantity, any significant correlations between rent control—both its presence and its strength—and the rental housing characteristics disappear. 13 It seems that most differences between rent control and non-rent control cities are due to disparities in income, racial proportions, and rental property market shares. Our models confirm most of the predicted theoretical relationships, with the exception of the key test variable—rent control—which exerted no significant effects, positive or negative. ...
> Rent control cannot be measured solely by its ability to keep rents low, but also by its protections from extreme rent increases and arbitrary evictions, and mechanisms for tenants to challenge unreasonable rent increases and landlord practices. Additionally, moderate rent control does give tenants greater stability, making tenancy closer to homeownership for those tenants with stable employment and incomes—in an area like New Jersey where homeownership is very expensive. Tenants under rent control have greater rights and security of tenure which perhaps makes rent controlled cities more desirable when choosing a city of residence