Why is my intuition wrong that one of two archers win in a tournament?

Take an example where the probability of Robin and Tuck hitting a target respectively are $0.9$ and $1.0$. Your method would give the probability of Robin winning being $\dfrac{0.9}{0.9+1.0}$ when the true probability is $0$.

Going back to your original probabilities of hitting of $0.45$ and $0.38$, you would do better saying the probability of Robin winning overall might be $\dfrac{0.45\times(1-0.38)}{ 0.45\times(1-0.38) + 0.38\times(1-0.45)}$ by looking at the decisive and mutually exclusive events of one hitting and the other not. This gives your $0.5717\ldots$

Whatever the probability distribution of the number of rounds, on any round Robin wins with probability $\propto r\bar t$ and loses with $\propto \bar rt$, hence the ratio

$$\frac{r\bar t}{r\bar t+\bar rt}=\frac{0.45\cdot0.62}{0.45\cdot0.62+0.55\cdot0.38}\approx0.572$$ as the game ends with probability $1$.