The ratio is of order $O(\ln r)$. This follows from the fact that the triangular truncation is bounded on the Schatten class $S^p$ (=the operators $A$ on $\ell^2$ such that $\|A\|_p:= (Tr (A^*A)^{p/2})^{1/p}$ is finite) and has norm $O(p)$ as $p \to \infty$. Indeed, for $A \in M_n$ of rank $r$ and operator norm $1$, then for all $p \geq 2$
$$ \|T \circ A\| \leq \|T \circ A\|_p \leq C p \|A\|_p \leq C p r^{1/p}.$$
I suffices to take $p = \ln r$.

The fact that the triangular truncation has norm $O(p)$ on $S^p$ is well-known. One reference that might contain a proof of this fact (or contain a reference) is the paper E. Davies, Lipschitz continuity of functions of operators in the Schatten classes, J. London Math. Soc. 37 (1988), 148–157. This is also related to the fact that the Hilbert transform is completely bounded on $L^p$ with norm $O(p)$ for $p \geq 2$.

**Edit** I had a look at the paper by Davies that I mention above, and indeed he uses that the triangular truncation $T$ is bounded on $S^p$ with norm $O(p)$. For this fact, he refers to a book by Krein and Gohberg *Theory and Applications of Volterra operators in Hilbert spaces*, where they attribute the result to Krein and Gohberg, *On the theory of triangular representations of nonselfadjoint operators* (1961). I am not sure I understand exactly how their result formally implies that $T$ is bounded, but here is a very simple proof that is probably their proof (and that is very close to the standard proof of the boundedness of the Hilbert transform on $L^p$).

Denote by $C_p$ the max of the norms of $T$ and $(1-T)$ on $S^p$ (actually it is not hard to see that $T$ and $(1-T)$ have the same norms, but this is of no use here). Then $C_2=1$, and if we prove that $C_{2p}\leq 2 C_p$ we have by interpolation that $C_p \leq p$ for all $p\geq 2$. To prove that $C_{2p} \leq C_p$, remark that for every $A$,
$$ (TA)^* (TA) = T( (TA)^* A) + (1-T)( A^* TA )$$
Hence
$$ \|TA\|_{2p}^2 = \| (TA)^* (TA) \|_p \leq \|T( (TA)^* A)\|_p + \|(1-T)( A^* TA )\|_p,$$ and each of the two terms is less than $C_p \|A^* TA\|_p \leq C_p \|A\|_{2p}\|TA\|_{2p}$ by Hölder's inequality. If we divide both sides by $\|TA\|_{2p}$ and take the supremum over $A$ we get the result.