So either they managed to make that more efficient somehow or you'll have exactly jack shit left after turning on wifi on it. Still, the ESP32 is not exactly cheap, but I have a feeling this $7 cost is the usual fantasy they advertise, then sell them for twice the price everywhere.
The current pico is sold at the stated price + taxes here in Sweden, so I don't see why the W wouldn't also be. And it seems unlike that Sweden would be special in that regard.
OTOH Zero is $5 only because it's effectively subsidized at various points in retail chain.
Pico W must be cutting it close at $6.
Edit: typo
The wifi stack doesn't run on the rp2040, the cypress module has its own cpus(2!) and ram to handle it. The lwIP stack runs on the pico but its resources requirements are essentially negligible.
So the rp2040 cores and memory are fully available unless you need TLS, which I'm sure will Absolutely steal both resources.