The site's wording fails for every $C\ge1$ at the set $A=\{1\}$: its reciprocal sum is $1\le C$, every $m$ is divisible by $1$, and nothing up to $x$ is left unsifted, so the answer is no for a reason that has nothing to do with sieving. The failure was observed in a thread comment by jif of 18 December 2025 (thread), which also noted the union bound $(1-C)x$ for $0<C<1$, and the site's commentary records it. The change inserts "with $1\notin A$" after "$A\subseteq[1,x]$", in the form of the condition that defines the quantity in the problem's own sources. The evidence, strongest first: Erdős and Ruzsa [ErRu80], p. 386, define the least unsifted count $H(x,K)$ for general sets over the $A$ subject to $\sum_{a\in A}1/a\le K$ and $1\notin A$ (display (1.5)) and announce for it the limit $e^{1-K}$ of $\log H(x,K)/\log x$ that answers this question; in Erdős's own [Er73], p. 135, the question is stated for $a_1<\cdots<a_k\le n$, and the coprime question printed next to it, display (14.3), takes $1<a_i\le n$; Ruzsa [Ru82] (display (1.3)) and Weingartner [We25] (the paper's condition (3)) define the quantity they estimate with $1$ excluded; and the site's commentary defines $H_C(x)$ as the minimum over subsets of $\{2,\ldots,\lfloor x\rfloor\}$ and says that the question asks whether $H_C(x)\gg x/(\log x)^{O_C(1)}$. The defect is already in [Er73], whose $a_1<\cdots<a_k\le n$ does not exclude $1$, and the site's wording keeps it; no source states the question as one about sets containing $1$. With $1$ excluded the recorded failure is removed, and the answer is decided by Ruzsa's and Weingartner's theorems rather than by the degenerate set. The observation about $A=\{1\}$ settles no instance of the corrected Statement and is credited here, not counted.