No, adjusting the exposure settings is equivalent to adjusting the brightness and contrast settings in an image editor. Your assertion would only have merit if the sun was impervious to exposure or contrast adjustments, which it is not impervious.
I finally got a sunny day, so as promised, here are two pictures I just took. One that has the Sun overexposed, and the second which has the correct in-camera adjustments.
I'm posting this to demonstrate that adjusting the cameras exposure settings is
NOT equivalent to adjusting the brightness and contrast settings in an image editor as Tom claims.
Feel free to load this image into Photoshop or Lightroom and adjust the brightness and contrast. You will never be able to turn the left image into the right image no matter what settings or adjustments you use. The data simply doesn't exist in the left side image, it's overexposed and no amount of image balancing can bring it back.
Conversely, you can't turn the right side image into the left side one by adjusting the brightness. You will never be able to make these two images look the same. The only way to make them is to adjust the camera settings when taking the pictures as I did.