A company that predominantly focuses on cost is a company that is going out of business.
There is a lowest level you can go with cost. If you cut costs too low, you limit your ability to make more revenue. You go down a deadly spiral till you are out of business.
The opposite is true, a company that focuses on revenue has no limits.
There is no known ceiling to how much revenue you can make.
In the theory of constraints, the focus is on increasing throughput, in lean it is flow.
Both methods look at getting the product out to the customer with quality sooner. This can result in more sales, better market response time and thus more revenue.
Flow or throughput is made faster by making the system more efficient. By increasing efficiency, you reduce waste, be it materials or effort. This results in reduce cost.
This means that reduced cost is a byproduct of the theory of constraints and lean. The first goal is getting a quality product out to the customer as efficiently as possible.
Speed of delivery is also a byproduct of lean and theory of constraints thinking.
By increasing efficiency, you reduce the manufacturing time, and therefore increase the tie of delivery.