I have seen the question here and have gone through the answer, but I still don't fully understand why the approach below, based on no-arbitrage, yields a different answer.

To summarize:

• At time $t_0$, I borrow $S_0$ cash and I immediately spend it to buy one unit of stock
• Whilst holding the stock, dividends are continuously compounded at a constant rate $q$ and reinvested into the stock until time $t$, at which point in time I will stop reinvesting and just take the cash, the value of which would be: $D_t=\int_{h=t_0}^{u=t}qS_udu$
• At time $t$, I need to repay the loaned money $S_0$, which has accumulated a continuously compounded constant interest rate $r$, i.e. $S_0e^{rt}$. I will also receive the cash from the counterparty for the forward (i.e. $F(t_0,t)$, which is the forward price agreed upon at time $t_0$), and I will need to deliver the 1 unit of stock to the counterparty, which I have held the whole time until $t$.

These transactions are summarized in the table below:

So for there to be no opportunity to make cash out of thin air, i.e. for the forward $F(t_0,t)$ to generate no arbitrage, we must trivially have:

$$-S_0e^{-rt}+F(t_0,t)+\mathbb{E}^Q_{t_0}[D_t]=0$$

i.e.

$$F(t_0,t)=S_0e^{rt}-\mathbb{E}^Q_{t_0}[D_t]=\\=S_0e^{rt}-\int_{h=t_0}^{u=t}q\mathbb{E}^Q_{t_0}[S_u]du=\\=S_0e^{rt}-\int_{h=t_0}^{u=t}qS_0e^{ru}du=\\=S_0e^{rt}-\frac{q}{r}S_0[e^{ru}]_0^t=\\=S_0e^{rt}-\frac{q}{r}S_0(e^{rt}-1)$$

This answer is obviously different to the one given in the linked question, so either the reasoning in my post is wrong (please point out where?) or the answer in the linked question is wrong (which seems unlikely, because the math checks out there).