Agile is designed for customers. Customers are the focus, and everything in Agile is designed to support delivering more value to the customer sooner than later; increased ability to realize a competitive advantage, be earlier to market - all of which translate to success for the business. It’s a win for everyone.
Agile isn't primarily about engineering. It is about making organizations more waste-repellent. Many companies have incorporated Agile practices into broader based business (non-development) practices, from strategic planning to customer services. Hence, it is too narrow-minded if you think "Agile is an engineering solution to an engineering problem, designed for developers, not managers." There is no conflict between true Agile and sound management practices. Well done Agile gives all stakeholders the data needed to predict when the scope for a project will be completed.
Besides managers seeking to be enlightened, the agility can be productivity multipliers by eliminating impediments: It's very important to understand the fundamentals of core agile principles which are not only in the spirit of Agile, but in a sense serve as a mature cornerstone of its implementations, they should also understand the fundamentals of creation of value, constraints, etc. Whether you have Agile teams or not, you still need to have management discipline and, if you prefer, leadership.
There is commonality between agile principle and leadership principle with three “I” focus: Interaction, Incrementalism, and Improvement. The principles of agile can be used by individuals and non-engineering groups as well. Agile helps management immensely through the power of collaboration, transparency, and feedback. Your mileage may vary depending on your degree of trust and discipline throughout your organization.
Agile is an effective way to break big business problems down into deliverable chunks to deliver value to the business at regular intervals. The role of the business product owner is critical in that you have a constant touch point on whether you are delivering what they want so that you can course correct as early as possible. It also inherently leads to better transparency to leadership enabling managers/leaders to apply levers to teams that are running behind or running into roadblocks.
Agile is a management philosophy. Agile is about considering and measuring human imperfection and the non-deterministic factors of a category of endeavors of which software is part of. Agile practice can help anyone or any group focus their attention effectively and take right action towards a vision in manageable segments where risk, uncertainty, unknowns and fast moving markets are involved.
Doing Agile is perhaps more as an engineering practice; but being agile is the ultimate goal organizations have to pursue, and business leaders/managers need to apply Agile principles and scale up agile practices from strategic making, to execution; from budget planning to customer success, it is a management philosophy to run a business and a methodology for problem-solving.