The only important part of 'general' is the ability to learn from experiential data and update your own model. That's what leads to general capability. Humans can't oneshot any task natively, but we can practice for a while until we uncover often novel methods of accomplishing something.
Therefore: the current transformer architecture is fundamentally incapable of AGI because the models have no mutable long-term memory.
You only have weights (large immutable memory), or context (small mutable memory).
Humans have mutable long-term memory: I can learn a new skill, adapt an old skill to new information, or learn new knowledge today that I couldn't perform/didn't know yesterday. I don't have a training cutoff.
Context engineering is an attempt to paper over this limitation. You can get really far with context engineering and huge models, but you will never get to AGI because there are many tasks where humans' mutable long-term memory outperforms.
For example, a human can invent a new musical instrument and then learn how to play the instrument they just invented. That's inference (inventing an instrument) leading to training (neuroplasticity). Humans have the ability to train our NNs with considerably fewer training samples. Everything that you can do with transformers is in one causal direction: training -> inference.