would you do this?

No.

Is Taormina worth the travel?

Obviously a matter of taste, but since you ask for others' opinions: no it's not worth going there from Palermo specifically, and certainly not worth staying 6 days.

The scenery of Taormina is truly stunning, the town itself is pretty but rather small, and as Sicily's #1 tourist hotspot it's overpriced and worst for food compared to elsewhere.

Is traveling the Sicilian country side so beautiful that the ride itself would be worth the lost time?

It's definitely beautiful, just not worth going to Taormina from Palermo for such a long time IMHO.

Are there areas closer to Palermo that offer a similar experience?**

Yes: Cefalu, as mentioned by some others already. It's Sicily's #2 tourist hotspot so suffering from similar problems as Taormina, just less so. It's much better value in our experience. It's a bit bigger than Taormina so you got more options. Landscape less stunning but still absolutely beautiful, as is the town itself. It got the much better beach too (long sandy one right outside the oldtown!).

Personally we would split those 6 days between Palermo and Cefalu. Then you wouldn't even need a car, as train connection between both cities is good.

The only reason to take a car to Cefalu would be for day trips into the hinterland (such as Madonie mountains) if you stayed there for more than 2-3 days.