There comes a point in automation where the DBAs job is less to do tasks, as to handle the occasional breakdowns of the automated processes that do the tasks. I have joined a team that is fairly developed down this path, which is nice. It has certainly helped hone my SSIS troubleshooting, that's for certain. But we have countless jobs and SSIS packages that take care of things. Rough count, over 80 SQL Server instances on probably around 50 servers, with 1300 or so databases...so automation does help!
It can cut both ways. You have to be VERY careful about changing anything because a dozen ancient automated processes may break. As a deterrent to foolhardy folks poking around in prod, this can be seen as a feature, rather than bug, though.
Many of the processes we could homebrew automation get taken care of by 3rd party products. Spotlight by Quest does a really good job for us, once you tweak it enough to avoid all the false positives and noise.