It is very dependant on the individual car and the state or local requirements. Some states or areas may have stricter regulations on the amount of pollutants the car can emit.
Running cats with no smog pump will eventually cause the cats to clog up & fail. High mileage engines burn oil that ends up coating the inside of the cats and clogs them. You may not see any evidence of oil burning, but it is still happening if you have a lot of miles on an engine.
The smog pump provides extra air to react with the exhaust and the catalyst and break down the pollutants into CO2 & water vapor. It shuts off at Wide Open Throttle so it doesn’t use any real power when you want all the power you can get. It just doesn’t make sense to disconnect it on a street driven car.