To get the calculation right, let's start with one parameter, $\beta_0$. It's important to get the summations in the right places, and I think that might have been your problem.
The GLM with log link and normal distribution has estimating equation
$$\sum_{i=1}^n (\exp\beta_0)(Y_i-\exp \beta_0)=0$$
The log-transformed Normal has estimating equation
$$\sum_{i=1}^n (\log Y_i-\beta_0)=0.$$
These aren't the same.
In the first one, the weight $\exp\beta_0$ is constant and we can divide through by it. The remaining equation is
$$\sum_{i=1}^n (Y_i-\exp \beta_0)=0$$
which is solved by $\exp\beta = \bar Y$.
The second one is solved by $\beta=\overline{\log Y}$, so $\exp\beta$ is the geometric mean of $Y$:
$$\exp\beta = \exp \left( \frac{1}{n}\sum_i \log Y_i\right).$$
These are importantly different. For a start, the second equation breaks down if any $Y$ is zero or negative, but the first one breaks down only if the mean of $Y$ is zero or negative. The first one models $Y$ as having a Normal distribution; the second models $\log Y$ as having a Normal distribution
Once you introduce non-trivial $X$, the weight term doesn't factor out of the first equation and it gives a weighted mean rather than a simple mean, but there's still the same difference between taking a mean of $Y$ and taking a mean of $\log Y$.
(One would more often compare linear regression on the logarithm and a GLM with log link and Gamma error. These are much more similar; in particular, they have the same range constraints on $Y$. They still aren't the same.)
