I think the "fast and loose" goes much deeper than using a few terms of a Taylor series approximation (more like ignoring non-convergence!). Because mathematicians have to go through the same process of shrugging off the formal rigor to think intuitively about mathematics. (Terry Tao wrote at length about this process)

I can understand the desire to appeal to a wide audience, but in this particular talk I couldn't identify a single bit of "projected" knowledge that one could hope to retain from the talk.