Are you still flying into LUX? Trains are free of charge in Luxembourg, so you can travel from L'burg city to the German border without tickets. Buses connect LUX with the station.
The car: Were you somehow turned off by the high train prices you might have found online? It's not easy to suss out those costs for comparison.
Generally speaking - 5 is the perfect number for traveling by train economically. Any day pass or multi-day pass you buy covers up to that many persons. So from the German border with Luxembourg to a Rhine town like Boppard or St Goar - or to Cochem on the Mosel - the Rheinland-Pfalz ticket (day pass) covers everyone in your family for €38.
https://www.vrminfo.de/en/tickets/tickets/leisure-tickets/rheinland-pfalz-ticket/
Then there's another day pass, the VRM Mini-group ticket. €30.50/day covers everyone on any trips you wish to make within the VRM travel zone (Which includes Cochem, Boppard, St Goar, and many other Rhine and Mosel towns.)
https://www.vrminfo.de/en/tickets/tickets/leisure-tickets/minigroup-ticket/
These passes can be bought on the spot. No reservations, no pre-purchase. Both passes allow free hop-off, hop-on travel and near-100% flexibility. And they cover any buses you might need (like the ones that go to Burg Eltz.) No insurance, no contracts, no environmental stickers, no parking fees... Travel on passes is vastly simpler than on train-specific tickets or traveling by car, IME.
Oh... almost forgot... Cheaper yet - free, actually - the VRM Guest Ticket, is another pass which every individual staying in BOPPARD gets for free - and it's valid throughout your stay in Boppard. It's good throughout the VRM zone - and a little further in some cases.
Cost isn't everything. People often choose a car despite the low cost of train travel without an understanding of the system, the assumption being that trains and schedules will somehow slow them down or hem them in. I've made uncountable journeys from several different base towns in the Rhine and Mosel Valleys by train. And I've done some by car too. But the fact is that trains go nearly everywhere here, and the system is extensive, frequent, and generally efficient. The only real trick is to base yourselves in a town with good connections to other places - and these rivers have several really good ones.
Besides the free guest ticket, Boppard, a great place on it's own merits, is the first place you should be looking for lodging on the Rhine. It has more lodging options than any other small Rhine town, it's a convenient travel base, and it's a convenient place to finish an afternoon river cruise as well.
Cochem is a good pick for the Mosel base.
There are many other towns on the Rhine and Mosel railways, however, that could work well for you as well. Some of the innkeepers in other towns offer the VRM Guest Ticket as well, but they are few in number. Boppard issues these passes universally.
From the Mosel there are connections to Paris (are you still headed there?) via Saarbruecken and Luxembourg. All you have to do is enter the town names. Cochem (Mosel) > Paris for example:
https://int.bahn.de/en/
I just did that and found a train with one change - the trip at 8:41 am takes 4 hrs. 11 minutes.
There is plenty to see and do in both river valleys, plenty that is not included in Rick's materials in addition to what he does cover. 6 days there is a good plan.