It's a legacy thing, basically all of them are considered "one-handed cutting weapons", so they are listed together as if it makes no difference (which is sort of true before the weapon specialization update). The upgrade system just picks the one nearest to the right as the "latest" weapon, and upgrades into that. Since burning axe is unlocked with a sword (if I remember correctly - I never use it) that is listed to the right of it, it can never be upgraded to. There's also a couple more examples of why the current system don't work well, like Ignis Bows upgrading into mundane bows (it's why I always skip the mundane bow techs), and Lightning Pikes upgrading into Pikes (why I always skip the last weapon tech). As it is, only a few things are worth using the upgrade system for, and even then, you usually have to plan accordingly before hand.

I suspect that they'll have to revamp the system sooner or later, it just hasn't happened yet.