Your GAS tour offers time on Day 2 for Burg Eltz and the Mosel River (a tributary to the Rhine River.) Oddly, there is no time at all allocated for the Rhine except for Koblenz itself, which straddles both rivers. The Upper Middle Rhine Valley is one of Germany's most popular destinations. UNESCO lists it as a World Heritage site.
https://whc.unesco.org/en/list/1066
The train ride between Koblenz and Bingen, which takes you through the rocky river gorge past villages, vineyards, and some 40 castles and castle ruins, is considered Germany's most scenic railway trip by Eurail and others.
https://www.seat61.com/places-of-interest/rails-down-the-rhine.htm
And day cruise boats cover the Bingen > St Goar segment intensely every day of the week for those who wish to spend a couple hours on the water.
So at a minimum I would suggest heading straight to Koblenz - or to one of the other Rhine towns listed below - for the purpose of just traveling through this part of the Rhine. To take this outing, ride the train south to Bingen, then cruise back to St Goar, and complete the round trip with a final train ride from St Goar to Koblenz. Area map:
http://www.loreley-info.com/eng/rhein-rhine/walking-hiking.php
The old-world towns of Rüdesheim, Bacharach, Oberwesel, St Goar, Braubach, and Boppard make for interesting hop-off stops as you proceed south by train from Koblenz. You might arrange a couple of hours in one or two of these places.
The K-D cruise schedule is finally out (in German.) Scroll down to the next-to-last page for the "Fahrplan Hauptsaison" (main season schedule) or the "Fahrplan Nebensaison" (shoulder season schedule.)
https://www.k-d.com/fileadmin/KD_Fahrplaene/KD_Fahrplan_2025_web.pdf