This is correct, it does apply to all forms of wage increases. I wouldn't say "instantly," but yes that's why cities are perpetually expensive despite constantly increasing productivity. They are highly productive ergo they have high wages ergo they can sustain high rents ergo they have high rents ergo they "are expensive" compared to adjacent markets.
Mentioned above but I'll put it here too for other readers: the reason rent is unique is that in high COL areas, it's driven primarily by the price of land which is has zero supply elasticity. Higher prices induce supply in all other forms of goods and services.
Regulations are causing the fact that housing prices follow purchasing power of potential buyers: it causes a shortage in supply, which means supply side can dictate prices. Prices will be what buyers can afford. Just wait for the next recession, housing prices will fall, even with the same regulations.
(It’s not clear why this only applies to rent and not other things people spend money on.)