I think what Jesper and Amrani wrote should clarify why taking the cofiber is not very appropriate in this context and what Qiaochu wrote clarifies why taking the fiber of the delooping is "better".

Let me try to complement the picture. I apologize in advance for the long answer, but since we already talked about it, I feel I may not have made my self clear enough.

The recognition principle tells us that the homotopy theory (say, relative category) of $n$-loop spaces is equivalent to that of pointed $(n-1)$-connected Kan complexes. When $n=1$, the functor in one direction is $\Omega$ and we can denote its inverse by $B$ as Qiaochu did.

A more rigid equivalence between these theories (say for $n=1$) is given by the Kan loop group $\mathbb{G}$ and $\bar{W}$; for $n>1$, you essentially apply these functors successively. What this means is that you can ridigify every $n$-fold loop space into a simplicial group in a functorial way. Of course, playing with sing and $|-|$ allows you to get even a topological group model. There is some care to take here though -- rigidification is a \textbf{model dependent} question: if you take ordinary group objects in the homotopy theory of spaces as presented by $Cat$ and Thomason equivalences (and the induced homotopy theory), you will not get something equivalent to group-like $\mathbb{E}_1$-algebras but rather the homotopy theory of crossed modules aka pointed connected $2$-types.

So let's say you're in $Top$ or $sSet$. What is a principle fibration structure on a map $E \to B$ of pointed connected spaces? well, up to (coherent) homotopy, it is the same as a group object $G$ and a map $B\to BG$ such that $E\to B\to BG$ is a homotopy fiber sequence in a prescribed way. To say this "model independently", you should say that it is a pointed map $B\to X$ for some pointed connected $X$ with the structure described above and call such a map an h-principle fibration (Nikolaus, Schreiber and Stevenson call it principle $\infty$-bundle).

Now let us come back to your question. Suppose you start with a map of $n$-fold loop spaces $\Omega^n f:\Omega^n E\to \Omega^n B$ (that is $E\to B$ is a map of pointed $(n-1)$-connected spaces). The homotopy quotient $\Omega^n B//\Omega^n E$, is then taken to be $hofib(\Omega^{n-1}E \to \Omega^{n-1} B)$ and as such carries a structure of an $(n-1)$ fold loop space. Note that if $E\to B$ was an h-principle fibration with structure $B\to X$, then $\Omega^n B//\Omega^n E\simeq \Omega^n X$ and the map $\Omega^n B\to \Omega^n B//\Omega^n E$ would be equivalent to an $n$-fold loop map by your initial prescribed structure.

You rightfully ask: how can we view $\Omega^n B//\Omega^n E$ as a (homotopy) Borel construction? Well, if we're in one of the "rigidifyable" homotopy theories of spaces, we could first convert $\Omega^n E\to \Omega^n B$ into a map of group objects $H \to G$ and then take the bar construction $Bar(H,G)=|Bar_\bullet(H,G)|$ which is evidently a model for the Borel construction $G\times_H EH$. But how do you do it without rigidifying? well, assuming we defined the homotopy quotient as the fiber, we can take the Cech nerve $C_\bullet^h(\Omega^n B\to \Omega^n B//\Omega^n E )$ (here I really mean the homotopy coherent version, e.g. apply the ordinary Cech nerve after you replaced the map to a fibration). A perhaps surprising thing is that as a simplicial space it is weakly equivalent to $Bar_\bullet(H,G)$ (i.e. there is a zig-zag of simplicial maps) -- this means in particular that in simplicial degree $k$ it is equivalent to a product $(\Omega^n E)^k \times \Omega^n B$ and not just a fiber product. So it is natural to call $C_\bullet^h(\Omega^n B\to \Omega^n B//\Omega^n E )$ the homotopy action groupoid and refer to its realization as a Borel construction of the initial map $\Omega^n E\to \Omega^n B$. This gives you the universal property you alluded to.

This last trick works more generally: if you start with an h-principle fibration $E\to B$ and you rigidify it (say in simplicial sets) to a principle fibration $G\to E'\to B'$ then the Cech nerve $C_\bullet^h(E\to B)$ would be equivalent to the bar construction $Bar_\bullet(G, E')$. (Edit) Note that this means that for every h-principal fibration $E\to B$, $C_\bullet^h(E\to B)$ is a Segal space (like the ordinary bar construction of a group action is). But there is an other Segal space around -- $C_\bullet^h(*\to X)$ -- which is in fact a group-like reduced Segal space (it is a "Segal group" that presents all the coherent multiplication of $\Omega X$). If we assume (for example) $B\to X$ was a fibration of Kan complexes with fiber $E$ then we get a map of Segal spaces $C_\bullet^h(E\to B)\to C_\bullet^h(*\to X)$. So we are lead to some kind of notion of a Segal space over a Segal group. The homotopy theory of these would be equivalent to that of spaces over the realization of the Segal group and thus gives you a model for coherent group actions. What's happening in the setting of a loop map $\Omega f:\Omega E\to \Omega B$ is that the loop space $\Omega E$ coherently acts on the space $\Omega B$ via the map $\Omega f$ (just like the domain of a group map acts on the underlying set of the codomain). The homotopy quotient of this coherent action is what we denoted $\Omega B//\Omega E$.

wrongthing to be doing here. $\endgroup$1more comment