Particularly if you're shooting RAW, you're correct that two-thirds of a stop's worth of bracketing doesn't get you that much. Two-thirds of a stop is something you can get in post-processing with minimal to no loss of information, particularly on a fairly low dynamic range scene where you get the exposure vaguely right in the first place.

However, try a more challenging scene for a camera: something like a sunset with a much larger dynamic range. In this case, it may well be that:

1. The camera doesn't get the central exposure you want right - it has no real way of knowing whether to expose for the dark foreground, the bright background or something else.
2. The dynamic range of the image just isn't something the sensor can capture.

For both these problems, bracketing may help - the first by just making it more likely that you can an exposure vaguely close to what you want, and the second by enabling you to do HDR or similar with the set of images. However, for both of these you'll want a larger bracketing amount than two-thirds of a stop - I'd start with two stops and work from there. But equally I wouldn't leave bracketing on all the time; just turn it on when you know it's going to be useful.

Of course, all this assumes you're using a large sensor camera; if you're using a phone camera with a tiny sensor and maybe only JPEG output, getting your exposure right in camera becomes a lot more important.