What is the advantage of this Monte Carlo approach over a typical numerical integration method (like Runge-Kutta)?

I was wondering the same thing, but near the end, the article discusses using statistical techniques to determine the standard error. In other words, you can easily get an idea of the accuracy of the result, which is harder with typical numerical integration techniques.

as i understand: numerical methods -> smooth out noise from sampling/floating point error/etc for methods that are analytically inspired that are computationally efficient where monte carlo -> computationally expensive brute force random sampling where you can improve accuracy by throwing more compute at the problem.

Numerical integration methods suffer from the “curse of dimensionality”: they require exponentially more points in higher dimensions. Monte Carlo integration methods have an error that is independent of dimension, so they scale much better.

See, for example, https://ww3.math.ucla.edu/camreport/cam98-19.pdf

Typical numerical methods are faster and way cheaper for the same level of accuracy in 1D, but it's trivial to integrate over a surface, volume, hypervolume, etc. with Monte Carlo methods.