Southern Visit of the Irish Republic

It's at least a two-hour drive from Shannon Airport to Anascaul. Assuming you're on an overnight flight from the US and not an experienced "wrong-side" driver, I question the sensibility of this. Galway is a lot closer and all motorway driving. Still, it's at least an hour. A lot of people go to Bunratty to see the castle and folk park there. It's very near the airport and would allow you to catch up to the time change and get used to the driving. Or this could be the time to visit the Burren.
I would stay longer than two nights in your "ancestral area."
If you want to avoid crowds, don't go to Kenmare. It's not a rural area. It's overrun with tourists. I see Sneem mentioned a lot. How about someplace on the ROK?
Cobh isn't on the way to Cork. OTOH, it is easily reached by train as a day trip. The train ride is short and more scenic than the drive (reportedly).
It's a LONG way from Wicklow to Galway. Even from the Wicklow Mountains to Galway. Athlone might be a good choice for somewhere in between. Not for hiking but for historic sights.
Driving and parking in cities can be a challenge (and, sometimes, expensive), so consider smaller towns and villages nearer to your hikes and stone circles. For example, since you expressed interest in the Drombeg Circle, consider staying in Skibbereen or Baltimore.

A cute little pub with an interesting history in the village of Annascaul is the South Pole Inn - good place for lunch or a pint. The pub was established by Tom Crean back in the 1920's - after he retired from the Royal Navy where he distinguished himself by participating in several Antarctic expeditions with Byrd and Shackleton. He was second officer on the ill-fated Shackleton expedition of 1914. The pub is a little museum featuring photos and artifacts from his long career. Worth a stop if you have the interest.
Just down the road from the village is Inch Beach - one of the prettiest expanses of sandy beach you'll find anywhere. It was featured in the old David Lean film "Ryan's Daughter" years ago.
We were frankly disappointed with the hikes in the Wicklow Mountains with the exception of the area in the immediate vicinity of Glendalough - lots of clear-cutting of the forest that rather detracted from the experience.
Suggest looking at the All Trails website and app for hiking trails in Ireland. It's served us well on our own trips - good descriptions of the various trails, their relative degree of difficulty, etc. The user reviews will provide reliable updates of trail conditions.
For a unique and decidedly off-the-beaten-tourist path hike consider going to the deserted famine village of Cummeengeera on the Beara peninsula ... not far from Eyeries. The trail goes up and over a gentle hill into a remote valley that was once home to about 20 families in the pre-famine era. About 1.5 miles each way as I recall. You'll need waterfroof boots for that one - part of the trail gets a little boggy near the top. Worth it though - it's a pretty little place and has a history of its own. Can google it to see if it might be of interest.

Thank you both!

Unfortunately, our lodging is not malleable with the one fortunate exception of Wicklow; we are now considering stopping/staying in the Birr region on the way to Galway & hiking Slieve Bloom instead.

Robert, thank you so much for the recommendations of the South Pole & the Beara Peninsula. Exactly what I'm looking for!

Sneem is a good suggestion. We stayed there 7 years ago. I’m sure it’s more polpular now than then but probably not nearly as potentially crowded as Kenmare.

For hikes on the Kerry peninsula, look at the Kerry Way. Although it’s a through hike you can break up parts of it as day hikes. Gap of Dunloe from Killarney NP to the Black Valley is good. Also look at the Bolus Barracks Loop. Probably my favorite hike on Kerry.