I'm wondering if you think Mainz might be a good place to recover
from jet lag after landing at Frankfurt.
Sure. Mainz is a great place for wandering around - the huge tangle of pedestrians-only streets (in pink on the linked map) along with several interesting sightseeing options, the small-city vibe, and the outdoor cafes around the Market Square all make for a pleasant way to reset your inner clock.
Mainz hugs the west bank of the Rhine about 20-25 minutes by direct train from FRA. If you catch the S-Bahn, the first stop after crossing the river is the Mainz Römisches Theater station. It goes without saying that there's a connection between this peculiar name and the ancient Roman theater which was unearthed there in recent years. Anyway, if you want to stay in/near the old town zone, the Zitadelle, the manicured Mainz City Park (Stadtpark) or the riverfront, this is your stop. I've used the Ibis (just a block away) a couple of times as well as the DJH hostel (a longer walk through the Stadtpark and the Volkspark, an "official" German hostel with private rooms and a very good one - or at least it was 20 years ago when I last stayed there!) Back then the Hof Ehrenfels, situated in a perfect old-town location just off the Augustinerstrasse, was in somewhat sad shape. But the word is that it's quite a nice place to stay now. Then there's the The Hyatt Regency on the waterfront and the Favorite Parkhotel, with an attractive location inside the Stadtpark, both of which have always been budget-busters for me.
The alternative hotel area is near the Mainz Hbf station. See that tunnel that lies just beyond the MRT station in the linked photo? Trains from the MRT station dive in there and emerge at the Hbf station just a few minutes later. This is not a bad are area to stay in - just a longer walk from Mainz's principal attractions, more traffic, not as pedestrian-friendly or interesting. There are lots of hotels here to choose from, from the modern B&B Hotel to pricier options. The Königshof tends to be a popular choice.
From Mainz Hbf you can catch a direct ICE train to Würzburg in the morning in just under 2 hours. Longer journeys with a change of train are also possible.
From the suburban Mainz Römisches Theater station, one change of train and 15 more minutes would be needed for the same trip.
If you were to instead stay at/near the Frankfurt Hbf station, your direct train to Würzburg would take just 1.5 hours. But IMHO the minimal additional required for a Mainz stay is worth it.
The 3rd alternative might work for you too... Skip both Mainz and Frankfurt and take the 1.5-hr. train ride straight to Würzburg after your flight, then do your unwinding/resetting thing there.