Standardization is optional for most cases of OLS regression. The story is a little different for multilevel models where centering really can impact the interpretability of your variables.
As you noted, there are certain benefits of standardized variables in OLS from a results interpretation perspective. Where I'm most familiar with the recommendation to standardize variables is when there is an interaction variable or when the predictors don't have a meaningful zero value. Since the excerpt mentions the intercept in particular, it may be that the recommendation for standardization is related to this latter point. The intercept of a regression will be the predicted y-value when x is equal to zero (the general idea is the same for multiple regression, but obviously there are more x-variables to consider). That intercept is fine regardless of whether variables are left untransformed, centered, or standardized. The transformation of the variables just helps shift the interpretation of the intercept (and coefficients) so that it is easier to look at the estimates and get a sense of what the equation is telling you. If the predictor is standardized, then the intercept becomes the expected y-value at the mean of the x variable. The same is true when the predictor is centered, but the advantage then of standardizing (dividing the mean-centered transformation by the standard deviation) is that the coefficient of the predictor is now in terms of standard deviation changes in the variable rather than unit changes.
Where you might start becoming concerned with the scale of the variables you're using in the regression is is if you're estimating multilevel models, expect that readers/users of your equation will interpret the magnitude of the coefficient as it's "importance", or if you're using Bayesian estimation (simplifying the scales of the variables usually makes for an easier-to-sample posterior space and makes prior specification a bit more streamlined). Other than that, and maybe a few other use cases, centering or scaling your variables is just to facilitate the interpretability of your final regression equation.