I would take the warnings about driving jet-lagged seriously.
And I value Rick's tips...
"On arrival, stay awake until an early local bedtime... Jet lag hates fresh air, daylight, and exercise. Your body may beg for sleep, but stand firm: Refuse."
But I doubt Rick is suggesting that you stay awake by attempting to operate dangerous machinery, or to navigate unfamiliar territory in an unfamiliar car in Europe's most aggressive driving environment.
I say let someone else drive. If you have to leave the airport to pick up a car to save the €60, you'll be using the train for that - won't you? (I knew you could do it.)
I would probably use the slower regional trains to Rothenburg. Are you in a rush to get there? If not, spend the extra 45 minutes. Your total train travel cost will be less than the €60 airport pick-up fee. Just follow the terminal signage to the Regionalbahnhof, buy a day pass, and hop on. You'll get some of that exercise Rick recommends when you change trains in Würzburg and Steinach, and probably in Frankfurt too.
Another good option is to shorten the train trip and spend the first afternoon and night in Würzburg - then pick up your car there the next morning. Würzburg is a nice place on the northern end of the Romantic Road, quite good for walking and sightseeing. FRA-W'burg takes only about 2.25 hours on the regional trains. (I have picked up in Würzburg in the past and found the procedure fairly trouble-free - and it's fairly easy to get out of town from there.)
Würzburg panorama
Würzburg must-sees
Würzburg Biergarten
Würzburg market square
If you take the train to Rothenburg... when you're done in Rothenburg use another day pass to reach your next destination (Nuremberg??) and pick up a car there.
"We have used Hertz twice in Europe (successfully) but are considering other companies."
There tend to be a lot of very unhappy rental car customers in Germany and at FRA in particular:
Budget/Avis
FRA Hertz thread
Sixt thread
Avis thread