Is the engineering manager expect to proactively find work for their team? Are they responsible to keep the work pipeline full for the next upcoming cycles?
This is a product/project management responsibility, and in some organizations or teams some or all of this can fall on the engineering manager side. Doing a lot of both at the same time is typically a crummy gig, as you don't have time to do both successfully for a right-sized team (you'll be either a bad manager, a bad product manager, or mediocre at both), Uou'll either be overworked or feel bad at your gig all the time.
Helping the team proactively scope and prioritize technical debt work, as opposed to product work, is almost always a critical function of managers.
I would say yes on two counts, one for his/her own growth/benefit and other for the team's benefit.
If there is more work in the pipeline it keeps the team less worried about what are the future
tasks they can focus on, and this becomes especially important if the company is going through
a rough phase. In addition additional work also gives opportunity for the team members to
work across different projects and rotate people who are driven and want to work on different
projects/technologies a chance, and to retain them. Naturaally, when the team benefits and
succeeds it helps the manager and improves his standing in the organiaztion.
However the ugly side of this is office politics, where some managers continuously try to expand
their turf undercutting other teams so that they manage bigger teams and reap rewards.
(say a bigger title).
Helping the team proactively scope and prioritize technical debt work, as opposed to product work, is almost always a critical function of managers.