If you use 'multimodal transformer' instead of LLM (which most SOTA models are), I don't think there's any reason why a transformer arch couldn't be trained to drive a car, in fact I'm sure that's what Tesla and co. are using in their cars right now.
I'm sure self-driving will become good enough to be commercially viable in the next couple years (with some limitations), that doesn't mean it's AGI.
If someone wants to claim that, say, GPT-5 is AGI, then it is on them to connect GPT-5 to a car control system and inputs and show that it can drive a car decently well. After all, it has consumed all of the literature on driving and physics ever produced, plus untold numbers of hours of video of people driving.
It would be a really silly thing to do, and probably there are engineering subletities as to why this would be a bad idea, but I don't see why you couldn't train a single model to do both.
The only difference between the two is training data the former lacks that the latter does so not a 'vast gulf'.
>And I see no proof whatsoever that we can, today, train a single model that can both write a play and drive a car.
You are not making a lot of sense here. You can have a model that does both. It's not some herculean task. it's literally just additional data in the training run. There are vision-language-action models tested on public roads.