"As we reviewed in Part 1, this is a product so the arithmetic mean of the elements is not meaningful"

This is simply not true, as the geometric mean is bounded from above by the arithmetic mean. That is, the maximum geometric mean is always given by the arithmetic mean and that's the highly relevant connection between the two.

See for example my paper 'Sharper Than Sharpe: A New Portfolio Measure' publicly available at SSRN https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4823752

Investors are faced with a dynamic multi-period decision problem in which they must consider not only the expected arithmetic mean of their portfolio, but also their expected return path in the form of one-period returns, as both will influence their expected terminal wealth.

The arithmetic mean as a descriptive measure of return is only applicable to a static one-period model (see Modern Portfolio Theory, CAPM, Sharpe Ratio), i.e. it represents always the cumulative or total return over the entire period under consideration.

The use of the arithmetic mean in the stochastic differential equation of the Wiener process is a mathematical necessity, since it reflects the static infinitesimally small static one-period time step dt via its mathematical description in a differential form.