Whenever I'm joining wood together, no matter what type of joint, I turn to wood glue.
"Wood glue" isn't actually one thing. It's essentially a marketing term or selling aid more than a description of a specific product.
For a traditionalist the words wood glue would actually call to mind hot hide glue and nothing else. And both epoxy and presumably polyurethane adhesive you also mention are heavily used worldwide to bond wood, in boat building for example, so they could just as easily be called wood glue, even though they're also very good as bonding other materials.
You're referring to what I usually call the conventional wood glues, which are types of PVA (see note bottom). This wasn't developed specifically for wood, it was just found to be particularly suited to it. The simplest type of white PVA sold as wood glue is no different from the white craft glue children use in schools and craftworkers use for cloth and paper projects. It's also used as a bonding aid for plaster, as a bookbinding adhesive and as an additive for concrete.
What is it about wood glue that makes it the best choice for common joinery?
A better question would be: why are PVAs the most common choice? How a person defines best is always going to be somewhat subjective as there's always a component of best for them.
So why are PVAs the most common? I'd have to say it's probably mostly about convenience: it comes at the usable consistency in a bottle, it's easy to apply, easy to spread, doesn't set too quickly, easy to clean up and non-toxic. And of course it's important that it can form very strong joints in wood (the saying "as strong as the wood itself" is not an exaggeration, as destructive testing demonstrates).
While there are still a few adhesives sold for use with wood (such as the Cascamite much favoured by old-school woodworkers in the UK) which can far exceed the strength of PVA, this strength is actually irrelevant in any furniture context since PVA joints can already be as strong or stronger than the material itself if properly formed.
Note: PVA-type glues are by no means one product with a single set of characteristics — there are essentially three classes, Type-I, Type-II and Type-III, with distinct differences (primarily in water resistance).