I think you can safely drop the rapier out, especially if you drop the large 2 handed axes and swords too. Thus there will be only a 1 die size difference (d6 vs d8) between the Dex and Str based users. Are there any large east Asia weapons from your goal time period that are equivalents to great swords and great axes?
As, for the general DnD state of the weapons:
Sure, but consider this: in D&D you have on the one hand the shortsword, a light finesse weapon, analogous to the Roman gladius and other ancient swords, and on the other hand you have the versatile longsword. There's a gap between the two that represents the majority of sword evolution in the medieval period: the spatha and its descendants, the migration period, knightly, and arming swords. They were neither light (because you couldn't readily wield two of them) nor versatile (because they weren't made to be wielded with two hands).
This really bothers me. If we are to consider it true that there is a weapon size between most short swords (gladius, xiphos) and the two handed swords i.e the long sword, (and there were as mentioned the spatha, migration period swords, knightly - arming swords), then there is not indeed just a gap in the inventory, but a damage die "constriction". Wouldn't then a more proper damage die-weapon proportion include something like this:
-daggers and small blades (up to 20-30cm/10in - d4 (light)
-short swords and curved blades (up to 50cm/20in - d6 (light)
-long(er) swords and scimitars (up to 80-90cm/35in) -d8
-true long swords/bastard swords (90cm+/35in+) -d10 versatile d12 (heavy)
-great swords (renaissance "pike breakers") -2d6 (two handed) (heavy)
???
I would like to have some kind of a penalty involved if a d10 LS was used single handed though. Not being able to dual wield one is not an deterrent enough for a shield user i.e.