Once the livestream is captured, you have two audiences to serve. The people who want the whole service, from announcements through worship through the message. And the people who only want the sermon, later in the week, on their commute.
Keep the full livestream in the Livestream Archive series. Do not trim it. Do not edit it. It is the record of the whole gathering, and someone in your congregation will want to rewatch worship or the announcements at some point.
For the sermon series, make a copy. Duplicate the Media item into the sermon Media series. Now you have two independent items pointing at the same source video.
Trim the copy down to just the message. Set a start point where the sermon begins and an end point where it ends. Subsplash respects those trim points on playback, so listeners jump straight to the teaching without scrubbing through 25 minutes of worship first.
The full service and the trimmed sermon now live in their proper homes. Both are searchable. Both are subscribable. Neither one clutters the other.