Definition: A quick win is one that occurs within the first three rounds. A slow win is one that occurs in round $4$ or later.
Notation: A sequence of letters involving $A, B, C$ indicates a sequence of winners. For instance $ABA$ means $A$ wins round one; $B$ wins round two; $A$ wins round three. Also something like $(ABC)^n$ means that there are $n$ repetitions of $ABC$ in the sequence of rounds. With this notation we will always take $n\ge 1$.
The initial sequence $ABC$ has probability $\frac14\cdot\frac12\cdot\frac14=\frac{1}{32}$. Consequently an initial sequence of the form $(ABC)^n$ has probability $\left(\frac{1}{32}\right)^n$.
Similarly, an initial sequence $(BCA)^n$ has probability $\left(\frac{9}{32}\right)^n$.
Wins for $A$: Quick wins for $A$ are $ABA$ and $ACA$. Slow wins for $A$ are of the forms $(ABC)^n ABA$ and $(BCA)^n A$.
Wins for $B$: The only quick win for $B$ is $BB$.
Slow wins for $B$ are of the forms $(ABC)^n B$ and $(BCA)^n BB$
Wins for $C$: Quick wins for $C$ are $ACC$ and $BCC$. Slow wins for $C$ are of the forms $(ABC)^n AC$ and $(BCA)^n BCC$.
Probability that $A$ wins is:
$$\begin{array}{ccccccc}
& P(ABA) & + & P(ACA) & + & \displaystyle\sum_{n=1}^\infty P[(ABC)^n ABA] & + & \displaystyle\sum_{n=1}^\infty P[(BCA)^n A]\\
=& \frac14\cdot\frac12\cdot\frac34 &+& \frac14\cdot\frac12\cdot\frac34 &+&
\displaystyle\sum_{n=1}^\infty \left[\left(\frac{1}{32}\right)^n\cdot \frac14\cdot \frac12 \cdot\frac34\right] &+& \displaystyle\sum_{n=1}^\infty \left[\left(\frac{9}{32}\right)^n\cdot \frac14\right]\\
=&\frac{3}{32} &+& \frac{3}{32} &+& \frac{1}{31}\cdot \frac{3}{32} &+& \frac{9}{23}\cdot\frac{1}{4} \\
=& \frac{6579}{22816} &\approx & 0.28835\end{array} $$
Note that going from the second to the third line of the preceding calculation, both summations are geometric series.
Similar calculations show that the probability that $B$ wins is given by
$$\frac{1557}{2852}\approx 0.54593$$
And the probability that $C$ wins is given by
$$\frac{3781}{22816}\approx 0.16572$$
I note that these values are quite similar to OP's Monte Carlo values.