Delta hedging for an American call option on a stock with a continuous dividend yield

Let the dividend yield be $$\delta$$ and $$C_u, C_d$$ and $$S_u, S_d$$ be the up and down values for the stock and the call respectively over the period $$\Delta t$$.

In Hull and all other resources I've looked at, the hedge ratio stays the same in this case as the no dividend yield case, i.e. $$\Delta = \frac{C_u - C_d}{S_u - S_d}$$ which confuses me because the payoff of owning one share of stock is actually $$S_u e^{\delta \Delta t}$$ or $$S_d e^{\delta \Delta t}$$ so I would presume that the hedge ratio should shift to be $$\frac{C_u - C_d}{S_u - S_d} \exp (-\delta \Delta t)$$ Why is this not the case?

My attempt at a "reasonable" explanation:

Over short $$\Delta t$$ we have $$\exp(\delta \Delta t) \approx (1+ \delta \Delta t)$$ so that the approximate payoffs from the stock are $$S_u + S \delta \Delta t$$ in the up position and $$S_d + S \delta \Delta t$$ in the down position, where $$S$$ is the initial price of the stock, asymptotically, so we may just take the denominator in the hedge ratio to be $$S_u + S \delta \Delta t - (S_d + S \delta \Delta t) = S_u - S_d$$ as usual.

If I may draw your attention the answer I gave on this post here: "risk neutral probability for stock with continuous dividend" There I explain how the binomial tree is set up originally, and there you can see why you simply work with $$U=e^{\sqrt{\Delta t}\sigma}$$ and $$D=U^{-1}$$ as a modeller's choice. The influences from the dividend yield (conveniently, continuous in your case) and the payoff type (American) are then captured in the expected price of the asset per time-step (must equal the forward price at that step) and in the value per node (discounted expectation in the European case or discounted expectation vs. immediate execution in the American case).
The dividend yield is already included when computing $$S_u$$ and $$S_d$$. Please check the formula (10.9) in the link below that gives u and d using risk free rate, dividend yield and volatiltiy: