With such good post production in editors like Adobe is there any real benefit in shooting with manual settings when you can change just about everything in any shot you take?
Your question is based upon an assumption that is not entirely correct: that you can change just about everything in any shot you take. You can't.
Things such as depth of field and shutter speed are set at the time the picture is taken. If, for example, the shutter speed is too slow to freeze a moving subject there is absolutely nothing you can do in post production to reduce the blur caused by the moving subject at a too slow shutter speed. The same is true of depth of field. If you take a photo with a very wide aperture that causes the background to blur, you can not recover the details of the background to make them look as sharp in post production as they would look had you used a narrow aperture for deeper depth of field.
Even among things you can correct in post production to one extent or another the cost of doing those changes in post rather than at the time you took the photo is a reduction in image quality. If you underexpose by several stops you can increase the exposure in post, but you do so at the price of loss of shadow detail, increased noise, and reduced dynamic range. If you overexpose when you take the photo the detail in the highlights that are totally saturated are lost and there is no way to recover them. Correcting for lens distortion means a reduction in absolute resolution. The same is true if many of the other post production adjustments you might need to make: they affect the quality of the finished product to one degree or another.
Both "Lew the traveler" and Michael Clark are correct. I will compliment there answers with another aspect - Time. It often takes time to make corrects in Post Production - and I'm fairly skilled in Photoshop. That is, I rather take the extra 15 seconds to get the settings right, in the camera, than spend the extra 15 minutes in Photoshop.
yes, because the desired depth of field or shutter speed may be be the one the camera selects. In post-processing one can't do too much about inadequate depth of field or too slow a shutter speed.
Just to throw out one more scenario where post processing isn't an answer: high contrast scenes. I may be taking a photo where getting a proper exposure on the foreground subject causes the background to blow out to white, with no detail recorded. With a flash (or several) I can brighten the foreground, allowing me to correctly expose the foreground and background, capturing all sorts of detail. I might also equalize exposure with things like neutral density filters, reflectors, or other kinds of lighting tools.