Once you start as an Engineering Manager, you will likely start getting more requests for work, estimates and other kinds of input than you can handle. When this happens, one technique you can use to help prioritize all of those requests is to think about whether or not you're a bottleneck, and if so, how impactful that is for others.
You can picture this as a circle of dependencies. Each circle impacts different numbers of people. It starts with knowing how you are blocking yourself; then how you're blocking a team member from moving forward; then how you're blocking your team from moving forward; then your colleagues; then your stakeholders; and finally the whole organization.
Starting with how you block yourself, think of a situation where you need to prioritize your team's backlog but you can't do that until you have the stories in the backlog or you meet with a product manager to set the priorities on backlog items. So, create the stories or meet with the PM first. If you can't work with a PM, decide if you can move forward on your own and let the PM know that you did and that you're willing to revisit the sorting. Unblocking yourself can be hard sometimes but knowing that you're doing it is half the battle. See this post with some approaches to help.
The next circle out is when you're blocking a team member from being able to move forward. Yes, this is only one person being blocked, but part of your job is to remove obstacles for your team so don't BE the obstacle! Stay in close touch with your team so that you know when this is happening; or, short of that, make sure you've built trust with your team so that they're comfortable letting you know when it's happening.
After that, the circle to care about is how you may be blocking your colleagues. There are innumerable ways to do this. It could be documentation, estimates, quarterly planning; almost anything. Blocking your colleagues could end up blocking more than just the individuals - it could also mean that you're blocking their teams or your department (planning, for example).
The key takeaway is that using this approach can help you prioritize what to do next. Looking at it as different circles of dependencies informs your next steps and helps keep you from being a bottleneck. When you've done this and you're still stuck as a bottleneck, it's time for ruthless prioritization.