You might be interested in the article by David A. Freedman on Friedman's urn. He reports a simple and intuitive proof due to Ornstein, which only uses the strong law of large numbers.
In his notation the urn contains $W_n$ white balls and $B_n$ black balls at time $n$, $a$ and $b$ have the same meaning as in the question.
D. Ornstein has obtained this very intuitive proof that $(W_n + B_n)^{-1}W_n$ converges to $1/2$ with probability 1 for $b > 0$. Suppose first $a > b$. If $0 \leq x \leq 1$ and $$\mathbb P\left\{\limsup \frac{W_n}{W_n + B_n} \leq x\right\} = 1,$$ by an easy variation of the Strong Law, with probability 1, in $N$ trials there will be at most $Nx + o(N)$ drawings of a white ball; so at least $N(1 - x) - o(N)$ drawings of black. Therefore, with probability 1, $\limsup (W_n+ B_n)^{-1}B_n$ is bounded above by $$\lim\limits_{N\to\infty}\frac{a[Nx + o(N)] + b[N(1 - x) - o(N)]}{N(a + b)}=\frac{b+(a-b)x}{a+b}.$$ Starting with $x = 1$ and iterating, $$\mathbb P\left\{\limsup \frac{W_n}{W_n + B_n} \leq \frac{1}{2}\right\} = 1$$ follows. Interchange white and black to complete the proof for $a > b$. If $a < b$, and $$\mathbb P\{\limsup (W_n + B_n)^{-1}W_n \leq x\} = 1,$$ then a similar argument shows $$\mathbb P\left\{\limsup\frac{B_n}{W_n + B_n} < \frac{a+(b-a)x}{a + b}\right\}=1$$ The argument proceeds as before, except both colors must be considered simultaneously.