Partial answer: elementary arithmetic transformations show that
$$S(n,1,1)=\sum_{1\le y\le n}\dfrac{1}{y}\sum_{d\mid y}\phi(d)\lfloor n/d\rfloor$$
which allows for much faster computation since it is essentially a single sum.
I didn't push the analysis further, but my guess is that $S(n,1,1)$ is
asymptotic to $Cn\log(n)^2$ (with a log squared), perhaps with $C=3/\pi^2=1/(2\zeta(2))$.