"Whirl" tours or "Blitz" tours (Rick Steves coined the "Rhine Blitz" tour with a cruise for a similarly short cruise segment!) do not necessarily care that you get the best experience - or may not even have given quality much thought. Sounds to me like "GET" is probably happy to squeeze folks on a short segment that is convenient for them, tell them they had a Rhine cruise, and then drag them somewhere else.
Since you are on the way to A'dam and have bags... I suggest ending the cruise in St. Goar (which gives you the 1.5-hr. cruise) - and then if you want to tour Rheinfels Castle - https://www.st-goar.de/en/rheinfels-castle/ - (why not??) you can leave your bags with the tourist office in St. Goar (which is on the way to the castle before you start the uphill part.) They will hold bags during open hours for visitors like you (no lockers at the small town train stations, unfortunately.) Then have lunch and leave a little later for A'dam.
If you are on a rail pass, the stopover is no problem. You should repost if you need advice for your train tickets to A'dam and getting in that stopover time you need on your ticket.
July or August - no difference really, high season, lots of visitors.
The right Bingen train station to use for boarding the cruise is called Bingen Rhein Stadt. You can walk from there to the dock with bags. Buy your cruise ticket at the dock in Bingen. If you go with KD, they will give you a 20% discount as long as you show the train ticket (or rail pass) you used to get to Bingen. The cruise fare will be about €16 with the discount.