Null hypothesis of regression = grand mean of data? This is a very simplistic question, but is the "grand mean of the data" another name for the null hypothesis of regression? This horizontal straight line (slope of zero) is what we are comparing our model to, to see if it's a better fit?

YES
But it depends on what you mean by "the null hypothesis of regression".
Software packages like R give an overall test of the regression. This is testing if any of the non-intercept parameters are nonzero.
$$
y = \beta_0 + \beta_1x_1 +\cdots+\beta_px_p + \epsilon\\
H_0: \beta_1=\cdots=\beta_p=0\\
H_a: H_0\text{ is false}
$$
When we do the test, we are comparing the regression equation above to an alternative that does not consider any predictors:
$$
y = \beta_0 +\epsilon
$$
If we fit this model, we fill find that $\hat\beta_0 = \bar y$, where $\bar y$ is the grand mean of all of the $y$ observations. That grand mean tends to go by the "pooled" mean of $y$ for the empirical setting or the "marginal" mean of $y$ for the theoretical setting (though "marginal" also works for the
Thus, there is a sense in which the null hypothesis of a regression has an equivalence with the grand mean of all $y$ observations.
