If you believe the conditional expectation of $Y$ given $xxx$ is truly linear, your approach is completely reasonable. Your approach is equivalent to using the identity link function. Let $g()$ be an inverse link function:
E[Y_i|X_i] = g(X_i\beta)
Then if $g_1(X_i\beta) = exp(X_i\beta)$, i.e under the log-link and $g_2(X_i\beta) = ...
Consider a simple example, where you are trying to model salary (Y) as a function of years of experience (X) for employees of a large company.
After exploring the data on a random sample of n = 50 employees of the company, you feel comfortable fitting a Gamma regression model to the (X,Y) data collected for the sample.
Your first model assumes that the log-...