Polyurethene glues are NOT gap filling in a structural manner and per-ounce, it is on a par with the better epoxies in cost (as in high). The foamy, open-cell structure that results when it is allowed to cure in the presence of air just doesn’t have any sheer strength of which to speak. When cured, polyurethene glues do not exhibit glue-line creep like PVA formulations (Titebond III is rumored to be the worst for creep).
As some have noted, the polyurethene glues are messy and the squeeze-out from them can be quite a bit of trouble to clean up.
That said, they are waterproof in a way similar to epoxy formulations. This is not to be confused with water resistance (type I, II or III) which stops short of retaining its bond when submerged for extended periods. For incidental water contact type II or III PVA glues are sufficient when applied correctly. That includes taking into account such things as temperature, possibility of creep, glue color, etc.
If it was simple, we would only have one sort of adhesive and never need to make a choice.
(Personal preference is for hide glue, hot or cold unless the situation just doesn’t warrant.)