When to grow an organization vertically vs horizontally?

I believe that a lot of good solutions to problems that a business is trying to solve comes from the rank-and-file. A technology company, for example, get some of its most innovative solutions from the engineers who are hired to build the company's products. When a company is at the stage where its focus is on building its products then it should focus on growing horizontally to maximize its efficiency. A good example is Google where they have been in the mode of building up their product. It makes sense that they have a fairly flat organizational hierarchy since the problem they're solving now is how to build a better search, how to monetize search, how to come up with new technical solutions, etc. Having a large set of talented engineer maximizes their chances of success.

At some point, however, the organization starts to move beyond just building the product. As the rank-and-file grows, a new problem emerges: organization. How does it organize all the ideas, thoughts and interactions between the different pieces? This is when a company needs to grow vertically. Managers are there to provide the necessary organization, structure and guidance so that the company can continue to maximize its potential. This also means that there is a max height to the organization at different stages since there is only so much organization that is needed and when that height is exceeded is when we see the high level of ineffective bureaucracy.

Sometimes I see an organization grow in the wrong direction because they fail to recognize the problem that it needs to solve. Instead of growing horizontally when trying to deliver products (This doesn't mean that throwing more ppl at a problem means it'll get solved faster or better. Growing horizontally could also mean empowering existing employees to come up with solutions), a company starts to insert multiple layers of management. This might be fine if it was trying to solve organizational problems but since it isn't these additional layers becomes more a hindrance then benefit.