Alternative interpretation of multiple regression coefficients? If we have a linear regression of the form
$$
Y = \beta_0 + \beta_1X_1 + \beta_2X_2
$$
is it valid to interpret the coefficient $\beta_1$ as the associated change in $Y$ when $X_1$ increases by a unit of 1, when $X_2=0$?

Just omit the last caveat and your interpretation is basically accurate, though I will add one small edit based on dipetkov's helpful comment:
$\beta_1$ is the associated change in the expected $Y$ when $X_1$ increases by a unit of 1.
This holds whatever the value of $X_2$ (conditional on the model being accurate, though this caveat is also covered by the addition of 'expected' to the definition). Variation in $X_2$ is irrelevant here because there is no interaction term in your model.
