There's no intrinsic problem with rebinding features, but, in actual practice, they do a couple of things that bother me.
For one, not every game engine can include rebinding. Point-and-click games don't have rebinders, yet some of them can work with JoyToKey (as long as the mouse movement isn't too demanding). If you have a library of games that all rely on the same profiler, it creates a more uniform experience.
Some rebinding menus are custom-designed around the game's aesthetic, which means extra labor for the artist.
There are ways to avoid that labor, since certain Unity games include a common input menu (you'll see it before the game starts) but developers find ways to make them needlessly convoluted. Go Kart Go! Ultra! (for example) is a really good-looking and well-programmed game. But, its Unity input menu contained way too many slots that read "x-axis" and "y-axis" and trying to figure out which slots are valuable (and which ones weren't) was a dreadful experience for me.
Rebinding menus tend to be less flexible than JoyToKey - there are quite a few of them, where you can't put an Action or Jump move on a D-pad. Developers also like to include both rebinders and generic impositions, which means that a controller will start working, even before I've gotten the chance to consent to a profile, and that has a way of making me feel like I don't have control (no pun intended).
It's true that JoyToKey is not a good-looking tool. It reminds me more of MS Excel than (say) video games, and that is a setback. However, some of the unpleasant characteristics of JoyToKey could disappear in a few years. The version of JoyToKey that's on conflexgames.com was specifically re-structured, to be easy to view in a living room.