I think the terminology we use may be letting us down.
Rather than talk about flat or horizontal organization, think it would be more useful to think in terms of encouraging freedom, autonomy and reducing bureaucracy while achieving {x} purpose.
How can one design for this?
To summarize, diffuse authority, have authority be ultimately responsible to the group (aka the employees as a whole), and avoid informal networks and concentration of resources by rotating responsibilities and equal access to equipment.
An employee-owned co-op with some or all of these systems in place could achieve the freedom and autonomy you seek. It has to be proved in practice, however. Many co-ops have more traditional authority structures still. Some bureaucracy is inevitable when designing a decision-making process for a large group.