"One of the keys to success that I have seen in my career is the ability to add value to the company, above and beyond what others do. I look for ways to improve situations, to make things run smoother, to build a solution that helps the company, or a person in the company. This makes me more valuable, and also less likely to be on anyone's layoff list."
This is all well and good and indeed, normally good advice, but there is a real downside to what you are suggesting if one works at one of those behemoth companies where people are deeply involved in "protecting their little kingdoms".
I had such an experience years ago in my career when working for one of America's largest defense contractors. I was stunned to find how much time was wasted by people protecting their little domains and making sure no one else exceeded their job description lest they should make others look lazy. In fact, they were lazy with very little initiative and there was a general attitude that with government dollars, no one needed to go beyond their defined roles and if they did, they were labeled as "brown-nosers" and largely shunned.
Since that time I have seen this dynamic a few more times, always in large companies. Places where people have odd titles and if you asked their boss "What does [person's name] really do?", the boss wouldn't really know for sure.
As much as I commend and agree with your advice, the reality is that in some companies (in my experience, always the big ones) what you are suggesting is actually an express ticket to being outsourced. In such a case, my advice to anyone would be find another job where you are not a cog in a huge wheel, but instead can apply this good advice presented, and stand out positively for the very same actions that in another setting, are considered negatively.
There's no such thing as dumb questions, only poorly thought-out answers...