You can start blasting cool air in your face almost immediately, and if the car is overheated you can roll down the windows and stand outside for a moment to purge the hot air. And if you only meant the first couple minutes of driving, I don't think that was very clear.

Remote start is cheap compared to the price of a car anyway, if it makes a difference.

Also the sun isn't significantly hotter in summer than in winter; it's not going to make you overheat.