Of course. Too silly.

I don't understand how $\mathcal{G}$ is a group? Shouldn't the inverse of a nondecreasing function be nonincreasing?

The paper Structure theorems for homotopy pushouts. I. Contractible pushouts. by John Klein (MR1490203) is all about contractible pushouts.

I know that Vandembroucq proved a uniqueness theorem for fiberwise localizations of unpointed functors in MR1923222

Very nice! Using the reduced cones allows you to eliminate the difference in topology between the half-open interval and the circle.

For sure $k\neq 1$, but all other $k$ are ok.

@ConnorMalin I've wondered the same thing about rationalization, where, as I understand it, Bousfield and Guggenheim work in a model category of simplicial sets which are intrinsically simply connected? As in, part of their structure is to have a single $0$-simplex (and maybe a single $1$-simplex?).

If all the digits of $2^n$ are even, then all of the digits of $2^{n-1}$ are no more than $4$...

Interestingly, though, $\Omega\Sigma X$ is the suspension of $\Omega X$ in the category of topological monoids. I wonder if there is a category-switching context in which there might be a more positive answer.