Barrel processors are a spectrum and GPUs are on one end of it. Yes, the classic canonical barrel processors (e.g. Tera architecture) more or less work as you outline. That is a 40 year old microarchitecture, they haven't been designed that way for decades.
Modern barrel processors implementations have complex microarchitectures that are much closer to a modern GPU in design. That is not accidental, the lineage is clearly there if you've worked on both. I will grant that vanishingly few people have ever seen or worked on a modern non-GPU barrel processor, since they are almost exclusively the domain of exotics built for government applications AFAICT.
Modern barrel processors implementations have complex microarchitectures that are much closer to a modern GPU in design. That is not accidental, the lineage is clearly there if you've worked on both. I will grant that vanishingly few people have ever seen or worked on a modern non-GPU barrel processor, since they are almost exclusively the domain of exotics built for government applications AFAICT.