The EM and PM are often in the room with you to support conversations with stakeholders/dependencies, absorb the brunt of cross-team communication, make tough prioritization calls, and to monitor the progress you report via the bug tracker and in weekly sprint planning sessions. The engineer is responsible for determining what part of the project to work on next, and how to spend each hour/day/week in the service of its completion. PMs/EMs are only going to challenge your plans if they're egregious, and it would be unwise to create a sprint plan that would require 100% of your time.
Most people have interviewing and mentoring to do, several previous projects that occasionally need their attention, code reviews to give, operational incidents to attend to, etc. even before you consider the undocumented tech debt fixes, refactors, and productivity investments that aren't necessarily even mentioned to PMs.
If you spend too much time on that, of course, you'll fail to complete your projects, stop getting interesting assignments, and have no hope of promotion. If you spend too little, however, your projects will turn out to be operational/maintenance nightmares that may even be worse for your reputation and prospects than vaporware.
Each engineer is their own architect and project manager, with the scale of the project tailored to the engineer's level. (You advance by earning and then succeeding at increasingly complex project assignments). Product managers help determine the vision of what the product ought to be; engineers get it there.
> Engineering managers assign projects to engineers
So engineers do only work on tasks handed down to them. We just did not agree on what "task" means.
Taking the bigger picture into account when designing a solution is simply good engineering practice. I don't think that's what the GP was talking about.