AMD GCN absolutely supports async compute[1]. Radeon cards for years would only make use of the ACEs in pure compute contexts, as OpenGL and DX11 had no concept of a secondary command queue and could not make use of them. This is a big part of the reason why Vulkan/DX12 require so much boilerplate to get a triangle rendered.
The PS3's SPU definitely counts as async compute especially with how it was used later in the console lifecycle[2] once people had time to familiarize themselves with it.
However, in the current gen consoles, you don't have to deal with a different ISA, command queuing, and shared memory between the GPU and CELL processor. You are only writing HLSL/GLSL/PSSL and setting up an aggressive amount of fencing to transition resources between readable and writable states within the GPU.
Indeed it didn't, and DX12 is extremely old tech. Consoles don't have async compute right now. (Edit: apart from the PS3).