There is no such concept as no time. It always gets down to priorities.
If you ask product managers what are they responsible for, you will get various answers depending on the character of that product manager and the company in which they are working. Different product managers in different companies handle different tasks and different responsibilities based on their beliefs and how the company sees them.
But there is one thing a product manager is always responsible for: prioritizing the roadmap and backlog. Product managers are masters in prioritizing the work of other people. They understand very well that there is never enough time and resources to develop everything they want and that if they will claim that everything is important someone else will define the priorities. They make hard decisions prioritizing the work of their colleagues in order to make sure the whole company is focused on the right things that make the proper impact.
However, when it gets down to the tasks of the product manager itself, suddenly too many product managers forget their craft. I see too many product managers claim that they are burdened by backlog management and don’t have time for doing strategy or other important things.
Just like the development team is never sufficient to complete everything you imagine for your product; a product manager can never do everything he should do in a 24/7 week.
As a product manager, you need to learn to drop things and let them get picked by others (or even not getting done). You need to make sure that you prioritize your own tasks just as you prioritize your development team tasks. It means you need to think which of the things you need to do will make the biggest impact on your team and product just like you do for features.
And if you are not sure how to start, because you are so focused on chasing the backlog, I suggest you learn how to make yourself redundant.
Priorities are everywhere. They even exist in your personal life. You might think you don’t have time to write a blog, but in reality, you just prioritized watching your favorite TV series over that.
Just like you do for your team, the only way to make sure you are doing the important things in your life and work that make the right impact is by prioritizing them properly.