An additional remark (too long for a comment) that might be of interest:

Given the finite-dimensional counterexamples in the answers by Iosif Pinelis and Pietro Majer it seems worthwhile to note that we can use those examples to construct a counterexample in infinite dimension which is "stronger" in the sense that, for fixed $T$ and $S$, there does not exist any $C \ge 0$ the satisfies the required inequality, but "weaker" in the sense that the operators involved are only positive semi-definite:

**Example.** There exist positive semi-definite operators $T$ and $S$ on the separable complex Hilbert space such that
$$
\sqrt{T^2 + S^2} \le C (S+T) \qquad (*)
$$
does not hold for any $C \ge 0$.

Indeed, for each $n \in \mathbb{N}$ there exist, by the other answers, operators $T_n,S_n$ on $\mathbb{C}^2$ such that $\sqrt{T_n^2 + S_n^2} \not\le n (S_n+T_n)$. Since $(*)$ is invariant under multiplication of both $T$ and $S$ with the same positive number, we may assume that $\|T_n\| \le 1$ and $\|S_n\| \le 1$ for each $n$.

Let us then consider the operators $T = \oplus_{n \in \mathbb{N}} T_n$ and $S = \oplus_{n \in \mathbb{N}} S_n$ on the Hilbert space $\ell^2(\mathbb{N}; \mathbb{C}^2)$; it follows that those two operators do not satisfy $(*)$ for any $C \ge 0$.