I think there are two issues. 1) is understanding what initiative does easily for weapons and why it should exist, and 2) is not letting initiative affect actions that it shouldn't such as spell casting and movement.
Most games end up solving 1), this fast/slow powerful/weak weapon problem and all the math related to it, by adding a simple derived stat called damage per second. Of course, in this game, we could call it DPT, damage per turn, and a turn being defined as 20 initiative points.
For example, a dagger with 6 damage and +6 initiative would have a DPT of 7.8 (6 damage * (26/20) attacks per turn), a shortsword with 7 damage and a +3 initiative would have a DPT of 8.05 (7 damage * (23/20) attacks per turn), and so forth. It's the calculation we try to do in our head but we never get down quite as well as the computer can figure it out.
This stat would allow the user to quickly tell from the interface if one weapon is better than another in terms of raw damage potential. DPT should include effects of accessories, enchantments, strength, and be balanced so that fast weapons generally had less damage because attacking more frequently is an advantage in itself. So maybe the necklace of ice would actually give +3 DPT and would scale to the faster or higher initiative weapons appropriately.
Magic items should also rarely affect initiative directly, as this can upset the DPT balance. Swinging a bit faster with a maul is so much better than swinging a bit faster with a dagger. Those items should be converted to +% to initiative after weapon modifiers.
Lastly, heavy armor should be -% to initiative after weapon modifiers. Heavy armor shouldn't be coupled with daggers; it doesn't make thematic sense.
Alongside DPT, there should be a field or two that displays secondary stats such critical hit chance and multiplier, armor piercing, proc abilities, etc. Some weapons will be lower DPT, but have higher crit or more amor piercing, or ranged, and that's the trade off.
On 2), I agree with a lot of the above, but let's keep it simple: magic should cost 20, and movement should cost 20. Attacking with a weapon would be the only thing modified by weapon initiative. It would be easy to see 3 initiative stats, too
That would end kiting, because a unit who moved backwards can almost always be chased at the same speed unless the backward moving unit had more total movement (which shouldn't be as much of a problem now). You could no longer kite someone just because you had a higher initiative.
Thus initiative would only play into DPT but not into spells cast per turn or quick running speeds (which really are the # of squares in movement).
Anyway, I'm mostly rehashing, but hopefully the above gives some simplicity in UI if not in description.
Elemental Forums
(A minute after posting I edited my previous post to explain why I want to "shift" initiative, still similar to your idea, isn't it?).