What really killed the CGNs (nuclear powered destroyers) was VLS (vertical launch system) and AEGIS - the sensor suite that newer Navy destroyers and cruisers are built around. CGNs were all built around magazine fed rail-style missile launchers, and sensors were all designed when state of the art was a transistor. Refitting ships to modern sensors like AEGIS and modern VLS was almost a complete rebuild. Even early Ticonderoga class conventionally powered cruisers (CGs) that had rail launchers were retired early because of the cost of refit to VLS.
Incidentally, VLS allows for much faster deployment of missiles, and doesn't require deck space for different launchers for different missiles, and a ship with a modern VLS have a huge rate-of-fire advantage on older rail launcher equipped ships.
That may have been the reason for mothballing the old ship designs, but why weren't the new ones nuclear powered? That seems to be independent of the missile launcher design.
* Role of the ship: why have an expensive ship that turns into a radioactive reef in a role when one of its missions is to "take a bullet" for a carrier?
Mainly for cost reasons, plus nuclear reactors also take up more space and can't accelerate quite as rapidly compared to modern gas turbines. If the Navy ever decides to build a new cruiser class to replace the old Ticonderogas then it might be worth considering nuclear propulsion, but realistically the budget just isn't there to make it happen. The Navy can barely afford to provide enough qualified crew for their existing few nuclear vessels.
Incidentally, VLS allows for much faster deployment of missiles, and doesn't require deck space for different launchers for different missiles, and a ship with a modern VLS have a huge rate-of-fire advantage on older rail launcher equipped ships.