Here is a general explanation why purely infinite $C^*$-algebras admit no tracial states: Non-existence Tracial states.
Is my following explanation for non existence of trace on Cuntz algebra $O_n$ (for any $n$) is also correct?
$O_n$ is simple, so if there exisrs a trace $\tau$ then it is faithful (as $\{x|\tau(x^*x)=0\}$ is a two-sided ideal if $\tau$ is a trace). Take one of the partial isometries generate $O_n$, then $\tau(1)=\tau(s^∗s)=\tau(ss^∗)$ but $1−ss^∗$ is positive and satisfies $\tau(1−ss^∗)=0$. Thus, by faithfulness of $\tau$ we get $ss^∗=1$ which is a contradiction for $n\geq 2$.
Yes, that's a very natural argument. Simpler and nicer than the one I gave on the answer you quoted.