The inequality you propose would imply that$$ |\operatorname{li}(n)-\pi(n)| \ll \frac{\sqrt{n}}{\log n}. $$On the other hand, Littlewood (1914) showed that this relation is false, in fact he proved that the left hand side divided by the right hand side is $\Omega(\log\log\log n)$.
So the inequality you propose is false.
Added 1. Littlewood's original paper is "Sur la distribution des nombres premiers", C. R. Acad. Sci. Paris, 158 (1914), 1869-1872. For a modern proof, see Theorem 15.11 in Montgomery-Vaughan's book "Multiplicative number theory I." (which is apparently the last result in the book).
Added 2. The OP asked in a comment how the inequality above follows from his proposed bound. Here are the details. We clearly have (for $n>100$)$$ \operatorname{li}(n)-\pi(n) = S_1+\frac{\pi(n^{1/2})}{2} + S_2,$$where$$S_1:=\operatorname{li}(n)-\sum_{k=1}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}-\log(2)-\dfrac{1}{2}$$$$S_2:=\sum_{k=3}^{\lfloor\log(n)\rfloor}\dfrac{\pi(n^{1/k})}{k}+\log(2)+\dfrac{1}{2}.$$The second expression $S_2$ is a sum of less than $\log n$ terms, each less than $n^{1/3}$. Hence, by the triangle inequality,$ |S_2|\leq n^{1/3}\log n \ll \sqrt{n}/\log n$.Using also the OP's conjecture $S_1\ll \sqrt{n}/\log n$ we get, again by the triangle inequality,$$ |\operatorname{li}(n)-\pi(n)|\leq |S_1|+|\pi(n^{1/2})|+|S_2|\ll \frac{\sqrt{n}}{\log n}.$$This, on the other hand, contradicts Littlewood's result from 1914.