When we are talking about Agile we emphasise best practices, continuous integration, automated tests, having iterations... But we mostly disregard the product itself.
We learned in the hard way, that minor mistakes in the beginning may turn into big troubles at the end. So we may say having a schedule is crucial. You should not stick to a plan blindly, of course - “sticking” is not agile way - but you need to have a vision for seeing beyond and a roadmap to make true that vision. In order to create this vision we should know the product in detail; what the product is, what is essential for the product , what is demanded from us. Undoubtedly product “awareness” is not enough by itself, being able to manage the product is the main issue.
Let’s say, you have the product and the vision and also a roadmap. So, you need a product owner who knows the product and its business consequences. Every project is performed within a time and budget constraint, which means the product owner should balance the demands from the clients; with practicability of these demands while considering time and budget. S/he is the main connection point of the business with the organization, that is why a product owner should able to say “no”. The product owner should prioritise the features within the scope of resource constraints.
Agile is not only a project management methodology but a multi-disciplinary philosophy. If you want to succeed with Agile, you should focus on product, learn how to manage it. Have a product owner who really knows the product and how to meet the needs of a client efficiently.