Traveling From Hamburg - Where to go from there?

Berlin in the summer can be beastly hot. I was there in 2018 (admittedly, a very hot summer all over northern Europe), and it was 100F. OTOH, there is a whole lot to see in Berlin, and it's easily accessible by train from Hamburg.
Here is a rabbit hole for you to go down, The Man in Seat 61's Guide to Train Travel in Germany.
https://www.seat61.com/train-travel-in-germany.htm
The Harz Mountains Steam Railway looks like fun.
I would say Hamburg to Berlin to Frankfurt is maybe too much for 9 days. I would choose a direction, east, west or south. I assume you're flying into and out of Hamburg, but if you can fly out of Berlin or Frankfurt instead, that could make a difference.
As between Berlin and Mosel Valley in July (and I've done both), I would choose Mosel every time. Take the train to Frankfurt Airport and rent a car. Drive up to Koblenz and then down the Mosel. Lots of all of your favorite activities there. Folks on the Forum like Cochem as a base, but it's too touristy for me. With a car, you can stay anywhere. I found a great AirBnB in Enkirch, which has virtually no tourists. I wanted to stay up the hill in Starkenburg, but my travel partner said it was too scary (she's afraid of heights). Views for days up there.
Be careful that you are not driving after wine tasting unless you're a dedicated spitter. Tolerance for driving after drinking is very low.

Give yourselves a day or two post-conference to see Hamburg itself. It wouild be a shame for your wife to spend four days in Hamburg and not get to see more than just the confernce hotel. Then head to Berlin for at least four days, including a day in Potsdam. I'll let others offer additional ideas.

I do think of Berlin as a huge undertaking, a go-go-go kind of place that fits in better with trips that are longer than yours. It will likely usurp half of your travel time if you go there.

Is it necessary for you to return to Hamburg for the return flight?

You do not have to stray far for what you seek. Lüneburg, around 100,000 residents (35 minutes away) and Bremen, around half a million residents (around 60 minutes away) are undeniably worthwhile places that make for good day outings from Hamburg. Lüneburg lies on that half-timbered house route you posted. You might wish to stay there a little longer to see these places. The travel writer I linked to above does a great job with these towns.

If you catch a train a little further south to Hannoversch Münden, 2.5-3 hours from Hamburg on the German Fairytale Route, you'll find several hundred buildings of the old-world half-timbered variety. Beautiful place. A few mostly relaxing days could be spent here. Sleeping Beauty's palace is a short bus ride (35 minutes) from Hann. Münden., and right next door is the Tierpark Sababurg, a family-oriented animal park with, among other things, a free-flight falconry shows (I find these mesmerizing, myself.) Also nearby: the university town of Goettingen, 40 mi from Hann. Münden by direct train. I've always thought of Goettingen as a classy town, one with few tourists that feels very German.

The Rhine/Mosel region is my favorite part of the country. There is no need to go to FRA airport to rent a car, if that's your choice, as cars are available in the area, but really - there is no need to rent a car if you prefer the train, as you stated.

As for untouristy towns: It may surprise you, but there are many of these that lie right on the Mosel Valley Railway and the Rhine Valley railway. Bullay is a great non-touristy Mosel wine town not far from Cochem on the main railway between the Rhine and Trier, good for a multi-day stay, great for short hiking jaunts, and it's a a rail hub as well. I've gotten off the train there and bedded down on at least 3 separate occasions. The very short Mosel Wine Railway (which takes you to Traben-Trarbach and other nice towns) meets the Mosel Valley Railway in Bullay:

https://www.alpenbahnen.net/assets/images/autogen/k-IMG_2375.png

Bullay, across from neighbor-town Alf: https://www.bahnbilder.de/1024/zugsuchspiel-blick-auf-moseltal-gesehen-597115.jpg

On the Rhine, check out Osterspai, south of Koblenz on the east riverbank. Beautiful spot next to the Rheinsteig hiking trail, hardly a tourist in sight.

Well, first off make sure you see Hamburg. In addition to the obvious Harbor, Fischmarkt, Reeperbahn, Binnenalster, Speicherstadt you should include the Deichstraße / Cremon / Nikolaifleth, Nikolai Church, Michel / Krameramtstuben, Haifischbar, Altonaer Balkon, maybe rent a canoe or SUP on the Außenalster, the Treppenviertel / beaches of Blankenese for starters.

For trips around Hamburg you should check out the following: Lübeck, Bremen, Glückstadt, Lüneburg, St. Peter-Ording, Altes Land, Heligoland, Haithabu / Schleswig, Schwerin. All should be reasonable without a car.

You can also do a fair amount of day hiking and walking in the Lüneburg Heath, along the Baltic coast, or even in places like the Holm Sand Dunes outside Hamburg. If you are willing to rent a car, you could also explore areas like East Frisia's hole in the wall gems like Greetsiel or niche towns like Verden an der Aller.

And YES YES YES to the Harz. You will not at all be sorry!

Here's another vote for the Harz. I was not able to get there on my last trip, but it's at the top of my destinations when I head to Germany next. It has mountains, hiking, small towns, a steam train, and so much to see and do. Plus it's not that far from Berlin, so no need to travel a long distance.

Enjoy your trip—it sounds like a wonderful opportunity!

"Any further south for this trip seems unreasonable." Hamburg to Würzberg ICE in 3.5 hours. Bavarian Mainfranken Wein region. Was du willst.
Not a fan of the Harz.

Thank you everyone!

G3rryCee & Russ - looks like we will avoid going too far east and head south and southwest coming out of Hamburg.

jkh & HowlinMad - definitely will make time to see the sites in Hamburg!

Thank you all for the great sites in Hamburg and Mosel/Rhine areas. Looks like it will make sense to fly home out of Frankfurt.