I often think what an Amiga for today would look like, something different from the computers we usually have. One with just a GPU-like processor doing everything, exploiting massive software parallelism instead of dedicated chips would fit this description.

I think you’d still need a CPU.

The way GPUs and CPUs work are so different. I’d be surprised if you could have a usable computer with just GPUs without turning it into a CPU or having it effectively be single threaded.

Kind of. Remember the RPis are brought up by their GPU before the ARM cores become active.