Indeed it would be nice to power the AC with solar power, but how does this help the city to cool down? You would cool down the houses, but the heat from the houses would remain in the city, it would even get hotter.
As others have pointed out, you might do better bouncing the incoming radiation back out of the city with white roofs. And still others say you can do a bit of both.
Also, you could definitely do well to reduce your roof albedo by switching to white/reflective roofs. This point has been made endlessly over the last 30-40 years. No one actually does it. People are actually installing solar panels.
To the contrary -- at least in NYC it's extremely common. The vast majority of "tenement" apartment buildings have silver roofs, for this reason.
Just fire up Google Maps in satellite view over Manhattan and see for yourself. White-looking roofs everywhere (but silver in reality).
New construction with roof decks tends to use things like white-ish floor squares as well. They're obviously much cooler to walk on in the summer.
(While, sadly, contrary to your assertion, virtually nobody is installing solar panels in NYC. Landlords don't seem very interested so far.)
https://flatroofdoc.com/silver-coating-protecting-a-rubber-r...
Elsewhere I find a claim that the aluminum paint used was a 20th century thing, but now white paint is preferred as being more reflective.
Los Angeles has building codes for cool roofs:
https://www.ladbs.org/docs/default-source/publications/ordin...