I don't think this will answer your direct question either, but it is still worth mentioning. The following fact was explained to me by a combination of Lusztig and Geordie Williamson, and it motivates why one should study (completions of) simply-laced Coxeter groups $(W',S')$ with $S'$ infinite.

There is a unique 2-sided cell $C$ in any Coxeter group $(W,S)$, consisting of all non-identity elements with a unique reduced expression. Any such element has a unique simple reflection in its right (resp. left) descent set. Within this cell, the left cells $C_s$ are parametrized by $s \in S$, and consist of all elements with a unique reduced expression and with right descent set $\{s\}$.

One can take the elements of $C_s$ and give them the structure of a graph, where $w$ is connected to $v$ if $w = tv$ for some $t \in S$. Each vertex can be labeled with the unique element of its left descent set. The resulting labeled graph does not depend on the choice of $s$.

One can view this graph as encoding a simply-laced Coxeter group $(W',S')$. Suppose that $S'$ is finite. For each $t \in S$, the reflections labeled by $t$ all mutually commute, and their product is an involution in $W'$. Together, these involutions generate a subgroup inside $W'$ which is isomorphic to the original Coxeter group $W$. In this way, any finite Coxeter group is canonically embedded inside a simply-laced one. Any Coxeter element of $W$ is sent to a Coxeter element of $W'$.

For example, this operation will produce the embedding of $H_4$ inside $E_8$. It will produce the embedding of a finite dihedral group $I_2(m)$ into $A_{m-1}$, sending each simple reflection in $I_2(m)$ to a product of every other simple reflection in $A_{m-1}$.

However, it may happen that $S'$ is infinite, and that the collection of vertices labeled by $t \in S$ is also infinite. In this case, one does not have an embedding of $W$ into $W'$, because each simple reflection would have to go to an infinite product. Though I have not seen it defined and don't know the literature, I imagine there is an embedding of $W$ into some suitable completion of $W'$.

For instance, this gives the embedding of $I_2(\infty)$ into $A_{\infty}$, sending each simple reflection in $I_2(\infty)$ to the product of every other simple reflection in $A_{\infty}$.

Anyway, the upshot of all this is that one could potentially study arbitrary Coxeter groups using only (completions of) simply-laced ones.

I don't think this will answer your direct question either, but it is still worth mentioning. The following fact was explained to me by a combination of Lusztig and Geordie Williamson, and it motivates why one should study (completions of) simply-laced Coxeter groups $(W',S')$ with $S'$ infinite.

There is a unique 2-sided cell $C$ in any Coxeter group $(W,S)$, consisting of all non-identity elements with a unique reduced expression. Any such element has a unique simple reflection in its right (resp. left) descent set. Within this cell, the left cells $C_s$ are parametrized by $s \in S$, and consist of all elements with a unique reduced expression and with right descent set $\{s\}$.

One can take the elements of $C_s$ and give them the structure of a graph, where $w$ is connected to $v$ if $w = tv$ for some $t \in S$. Each vertex can be labeled with the unique element of its left descent set. The resulting labeled graph does not depend on the choice of $s$.

One can view this graph as encoding a simply-laced Coxeter group $(W',S')$. Suppose that $S'$ is finite. For each $t \in S$, the reflections labeled by $t$ all mutually commute, and their product is an involution in $W'$. Together, these involutions generate a subgroup inside $W'$ which is isomorphic to the original Coxeter group $W$. In this way, any finite Coxeter group is canonically embedded inside a simply-laced one. Any Coxeter element of $W$ is sent to a Coxeter element of $W'$.

For example, this operation will produce the embedding of $H_4$ inside $E_8$. It will produce the embedding of a finite dihedral group $I_2(m)$ into $A_{m-1}$, sending each simple reflection in $I_2(m)$ to a product of every other simple reflection in $A_{m-1}$.

However, it may happen that $S'$ is infinite, and that the collection of vertices labeled by $t \in S$ is also infinite. In this case, one does not have an embedding of $W$ into $W'$, because each simple reflection would have to go to an infinite product. Though I have not seen it defined and don't know the literature, I imagine there is an embedding of $W$ into some suitable completion of $W'$.

For instance, this gives the embedding of $I_2(\infty)$ into $A_{\infty}$, sending each simple reflection in $I_2(\infty)$ to the product of every other simple reflection in $A_{\infty}$.

Anyway, the upshot of all this is that one could potentially study arbitrary Coxeter groups (with $S$ finite) using only (completions of) simply-laced ones (with $S'$ possibly infinite).