One point that seemed to be missed in the answers so far (though I'll admit I didn't read every word) is that indexed shifters are only a small part of indexed shifting.
It used to be that you needed friction shifters because, in order to shift to a larger sprocket, you had to "over-shift" substantially -- push the lever beyond the point where it would eventually end up, then move it back once the chain had begun to move. Because of this, indexed shifting was essentially impossible, even though it was no doubt a sought-after goal of many inventors.
What changed was the chain and sprockets, with carefully-engineered profiles that would cause the chain to "climb" the sprocket if it was pushed ever so slightly in that direction. This invention made indexed shifting possible, and largely eliminated shifting as a major roadblock in the way of a "mass market" for bikes.
But, rather incidentally, the change to chain and sprocket profiles made possible shifting under load, something that was largely impossible before. While many of us would willingly give up indexed shifting (I miss the sensation of the lever that allowed me to inherently know what gear I am in), how many would be happy to give up being able to shift under load?