Let $\mathscr{C}_1$ be the little 1-cubes operad. If $X$ is an algebra over $\mathscr{C}_1$, I can think of (at least) two ways how to deloop it:

1. I can consider its two-sided bar construction $B_\bullet(\Sigma,\mathscr{C}_1,X)$, which is a simplicial space, and take its geometric realisation.

2. I take a ‘hands-on’ rectification to turn $X$ into a topological monoid, which generalises Moore loop spaces: Consider 
$$X':=\{(x,t)\in X\times \mathbb{R}_{\ge 0}\mid t=0\Rightarrow x=*\}$$

   with multiplication $(x,t)\cdot (x',t')=(c_{t,t'}(x,x'),t+t')$, where $c_{t,t'}\in\mathscr{C}_1(2)$ is the configuration of the two intervals $[0,\frac{t}{t+t'}]$ and $[\frac{t}{t+t'},1]$. This is a topological monoid and I take the realisation of its nerve.

I think it should be rather classical (or obviously false) that, under mild assumptions, $BX$ and $BX'$ are equivalent, even though the obvious projection $X'\to X$ is not a map of $\mathscr{C}_1$-algebras. I was hoping to find it in Fiedorowicz’s work on ‘Classifying Spaces of Topological Monoids and Categories’, and indeed, his Corollary 7.9 at least tells me that for $BX'$, it does not matter whether I take the two-sided bar construction or the nerve. However, this Corollary starts with a topological monoid, so it does not solve the question in full generality.

Does someone know a reference for this? Or an easy argument?