Polyurethane adhesives, whatever the brand, are generally very good adhesives. Used correctly they form a very strong bond. People who report they can see the glue line after using this adhesive type, or any other type for that matter, and note that the adhesive has failed are not actually apportioning fault to the right cause. It's not the adhesive that's at fault in that situation, it's the poor joinery that is the cause of failure. I'm not sure that what you describe is a sign of this type of fault, i.e., gappy joinery, but you also describe foam covering the surfaces, presumably this being the surfaces that are really should be intimately touching each other rather than loose.
Most adhesives can't structurally bridge a gap with the exceptions being epoxy resin formulations (often incorporating fillers) and to some extent urea formaldehyde adhesive types, e.g., Cascamite, which you asked about. Interestingly, many have noted they've experienced problems with the Polyvine branded Cascamite in recent months and years, seemingly largely centred around the stuff not mixing properly in the first place leading to imperfect curing. I too have had some problems within the last three or four years with this brand of urea formaldehyde adhesive, but I've had no need to use this type of adhesive recently, so I can't comment on other brands of this glue type .
Turning to talking in generalities, there isn't a single glue type that satisfies all the gluing needs of a woodworker, but one type that readily satisfies most needs is probably one of the PVA or aliphatic resins, but even this type varies in working properties. Some grab very quickly (Everbuild D4, for example) giving little time to align parts and clamp up effectively, and others give longer to get clamps on and to make sure everything is lined up properly. Some PVA formulations are water resistant or said to to be 'water proof', and others are not, so choosing which one to use depends on the end use of the project. In general, the more 'water proof' the PVA type, the quicker it grabs, therefore less time to assemble and clamp up your project. PVA also suffers from creep, and if this is undesirable in a glue up, e.g., in the case of tightly bent laminations, then another adhesive might be better, e.g., urea formaldehyde or epoxy resin, (both of which have to be mixed which takes time) and in the case of the latter adhesive there are versions that can give you up to an hour after its application to get everything aligned and clamped up properly - often useful in complex assembles such as bent laminations. If the wood is wet, i.e., by definition wood that has 20% MC or more, then a good option might be a polyurethane adhesive (some grab quickly, and others slower) because this formulation will bond wet wood as well as bonding odd materials together, whereas none of the other common types are formulated to work with wet wood, so will fail.
Others have mentioned hide glue, fish glue, and so on, and all adhesives have their uses, strengths and weaknesses, and choosing an adhesive requires matching its properties to the task in hand and end use of the project. But, as I said earlier, a good general purpose adhesive that covers many needs and uses will, for most woodworkers, be one of the PVA or aliphatic resins. I think it's probably true to say that 99.5% of woodworkers always have some of this stuff to hand ready to use at a moment's notice. Slainte.