I think you may underestimate the nature of exponential positive feedback. The AI singularity centers around an inflection point of self-programming before which noticeable improvements take place over months and weeks, and after which they take place over seconds and microseconds. Self-modification iterates faster than you can record.
It has nothing to do with “inspiration” or “actual intelligence”. It is entirely based on self-modification, and the “illusion” of intelligence is sufficient for that task. Eventually, the illusion is indiscernible from reality (spoken as a very complex method of distributing gametes).
Venn Diagram Rule (media.kbin.social)