I don't know what aircraft you flew on over 3 years ago. The current options from Etihad are Boeing 777 and Airbus A380. The route doesn't seem to be nonstop anymore, aboard any airline.
Along the shortest distance -- great circle route -- there aren't many options for an A380 to land and find support equipment mid-route (shown below). Refueling can occur with the passengers on board, there is a procedure to be followed, which includes alerting the fire service at the airport to be ready.
If we take only the cost of fuel into account, and a 777-200 as a baseline, actually stopping every ~6 hours is more economical. (This doesn't take into account landing fees, accrued cycles*, passenger convenience, etc.) Not just for your flight, but any long-haul flight; the fuel burned to carry the fuel for long distances becomes considerable after a certain range.
Another consideration is which freedoms of the air, say Etihad/UAE, has with say Russia. The second freedom specifically is:
The right to refuel or carry out maintenance in a foreign country without embarking or disembarking passengers or cargo.
While they are called freedoms, not all country-pairs mutually allow all the freedoms. The specifics are usually not public and are contained in the air service agreement between any 2 nations.
In all, it might have saved fuel even if we included a detour off the great circle. It certainly would have increased the maintenance cost. And it would have been an operational nightmare to arrange compared to a hotel and meal.
* Pressurizing and depressurizing, applying the takeoff thrust twice, applying the brakes twice, etc., all increase the maintenance cost.
(gcmap.com)