Without any adjustments to your itinerary, I penciled out fares for two adults doing the whole trip by train.
For the Black Forest, I assumed a couple of free day trips on the scenic Black Forest Railway, and/or perhaps on the Hell's Valley route.
I used the 4-day sale-priced (€288/2) German Rail twin pass (valid 1 month) currently available at DB for 4 of your longer routes...
1.) FRA-Erfurt 2.) Dresden-Berlin 3.) Wittenberg-Cologne 4.) Mosel Valley's Cochem-BadenBaden
10 of the 11 additional segments, which I "divined" based on your rough sequence, can be done on day passes of one sort of another ranging in price from €20 to €52, averaging €31/day. Total = €311
Together, that's €599.
Then you have a final long travel segment for Füssen - FRA, for which you need a saver fare ticket at some unknown price.
THE KIDS: If they're under 12, they travel free on that saver fare ticket to FRA. If they're under 15, they travel free on the day passes; on some day passes they can be older and still travel for free.
Assuming you intend to KEEP all your destinations, if I were you, I would look at reversing your route in Bavaria at a minimum, if possible... Black Forest-RR-Nuremberg takes you to the northeast, then you plunge south, then return north to Frankfurt. I'm guessing wildly that the kids must meet you in Nuremberg - otherwise the more logical route would be Black Forest - Füssen then the rest of Bavaria and north. That would eliminate the long final journey. Würzburg might be a FINAL stop since it's only 1.5 hours by train from FRA airport.
You will likely have some additional day trips here and there that neither you nor I have accounted for, of course.
Driving the entire Romantic Road, if that was your plan, is possible but probably not necessary. If you really WANT to do that, you might rent a car for a couple of days. But most of the towns can be visited by train.
Your impulse to "...rent a car to do the obvious driving parts (rivers, Black Forest)" assumes the absence of railway service in these places but in fact Germany's river valleys are full of trains. They line both sides of the Rhine River, connecting all the towns there. Most of the popular stops on the Mosel - Cochem, Traben-Trarbach, Trier, Winningen, for example - are connected by train as well, with the occasional bus connection or boat ride (as in the case of Beilstein and Bernkastel.)
Mosel Rail Map: https://www.bahn.de/moseltalbahn/view/freizeit/moselsteig-karte.shtml
The Black Forest train routes tend to follow the local rivers too. See BF rail lines on this map. The same is true for towns in the Main River Valley and the Neckar River Valley, which you haven't included, and others as well.
Germany mountain towns have train service too. Map of Garmisch region:
https://bahnland-bayern.de/assets/images/4/BEG_GR_Regio-Ticket_Werdenfels_2016-09-Ansicht-6e09290c.jpg
That part of the above train route through Austria (Außerfernbahn, very scenic) also provides access to Füssen (w/ bus connection to Reutte) and on the "Fürstenweg" trail to the Füssen area palaces from the Ulrichsbücke stop:
https://en.wikipedia.org/wiki/Ausserfern_Railway#/media/File:Au%C3%9Ferfernbahn.png
Train between Munich and Füssen:
https://www.youtube.com/watch?v=H7ckkP_nUR0