Also, a few mechanical thoughts.
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I like the idea of 'roll the damage die as your initiative'. The thing is, it *does* mean you are tied to using *that* weapon when your turn arrives, and the 5E concept of a 'free' weapon draw before acting gets nerfed. So, I can see why all the melee rolls are the same. It preserves 5E action economy and closes a loophole of (rolls d4 for dagger) (draws and attacks with greatsword).
Maybe if you declare you attack with a weapon, you choose which one, and get your free draw when you roll initiative?
But, 5E does try to create a lot of fluidity and eliminate the 'tactical square-based board game' feel of 3E and 4E. People can move, draw, attack, move. So, again, back to just having one roll for all melee weapons, even if that does gloss over the whole 'dagger vs. pike' everyone argued about in 2E.
Then again, after having witnessed and participated in a lot of different things like Kendo, SCA combat, martial arts, etc. the idea of different weapons having different speeds is a bit less cut and dried as you might imagine, so IMO the abstraction level of combat can take that into account.
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If a ranged attack gets a smaller die than a melee attack, then any ranged attack that would be made at disadvantage in combat - i.e. a short bow user with a sword wielder in their face - means the initiative roll should be made at disadvantage too.
But, again, more complexity.
It's tempting fix it by going back to the idea of 'size of damage dice is your initiative', so longbows would actually be more unwieldy than daggers, short bows would have a functional advantage vs. long bows for once, etcetera, but the 'free draw and attack' issue is still bad.
What if melee ranged attacks rolled like melee (d6), and bow/crossbow attacks rolled like spells (d10)?
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I think some things might be simplified a bit, too.
Like, the Dominate spell issue. If you control someone, they act on your turn, after you. Period. No dice rolling.
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I think to make this really fly, I'd like to set up an on-table initiative board. Slots numbered 1-20, and everyone has a unique counter for their player (maybe even two of them, so they have one they hold for reference and one on the board), and they just put their counter on the slot with the right number.
Stuff like delaying would be faster to figure out.
My players are all avid board gamers, so adding a prop like this to the table would go over well. They're not tied to 'this is NOT MY D&D' and are wiling to try new things. I think I'll try it in my game when I beta test this system.