It's difficult to work in a horizontal organization if you are still thinking vertically.
Why? You will miss opportunities, primarily. But, it will also appear confusing as people take on responsibilities outside of what may be written in their job description. It may seem to others as though you aren't focusing on your job as your role within the organization expands.
In order for a horizontal organization to work well, you need to have people that are willing to take on multiple roles. Otherwise you will isolate yourself and have nothing to offer others.
When you limit your involvement in the organization, you limit your potential to expand. Notice that I didn't say "move up." The action of moving up to a more senior role happens in vertical organizations. In the horizontal world, there is only expansion or contraction.
I want to work with people that are capable of handling multiple functions within an organization. This is not unlike how a CEO works on a day to day basis, so why should the employees function any differently? This is important to a company because it enables them to more with less. A company doesn't have to go out and buy a resource--they can develop it from within.
Yes, there will be people that aren't interested in doing anymore or less than what is specifically required of them. And that is ok. As a leader, I want to help people get where they need to go. If that means people are engaging at certain times, and not at other times, then that is fine. Managing that balance across all of the people on your team is one of the unique challenges facing leaders.