Interpretting coefficients of a GLM as multiplicative

I'm trying to make sense of the coefficients of a generalized linear model with negative binomial error structure, and a log-link function.

From Gelman and Hill, "Data Analysis Using Regression and Multilevel/Hierarchical Models", pg 111:

"The coefficients $\beta$ can be exponentiated and treated as multiplicative effects."

Such that if $\beta_1 = 0.012$, then "the expected multiplicative increase is $exp(0.012) = 1.012$, or a 1.2% positive difference ...".

My question is how do you interpret these coefficients in a model with many covariates, specifically where other coefficients have enormous standard error (i.e. non-informative/significant)?

It seems like interpreting the coefficients as multiplicative isn't as intuitive then. For example: if $\beta_1 = 10$ (and $exp(10) = 22026.47$), and the standard error is 30, then the other coefficients in the model are multiplying this huge percentage! Is it better to not interpret the coefficients as multiplicative in this context?

Any thoughts are appreciated!

Second, if you have such a large coefficient then the corresponding regressor typically takes rather small values where a 1-unit change in the regressor is not realistic. Instead, imagine a 0.01-unit change might be realistic: Then the multiplicative effect of that would be $\exp(10 \cdot 0.01) \approx 1.105$, i.e., corresponding to a 10.5% increase.