Y'all have diverged a lot from the discussion idea of "removing the to-hit roll".
I feel like basing melee off the combatants' comparative skill level makes sense, while ranged combat is based almost entirely on the shooter's skill level (with maybe a little modifier for the target's dodging activity). But even there I look at "compare the skills involved" as "make a to hit roll", just dressed up differently. And if the "roll" involved is based on skill, then you minimize the potential importance of the weapon. If "difference between my attack and your defense is base damage, plus X" (on d20s), then either the X has to be huge to be meaningful, or the difference between a dagger and a greataxe is easily overwhelmed by a wide spread on the dice.
The other ideas I've seen mentioned on here - like rolling multiple kinds of dice, based on skill and weapon and (insert other factor here) might make that a little different, though. If your skill gives you two to four d6s, and then you add the weapon's damage die as well (or a "damage die" based on the defender's shield), then I could see weapons having a larger impact... and daggers never being used.
I like the idea that a dagger can be as deadly as a greataxe in the skilled combatant's hands, but I also like there being a good reason to use different weapons (which 5e doesn't have, admittedly, but 3.5e did). Finding/building a combat system that makes both skill and weapon choice impactful without bogging things down though - especially for that one player at each table that takes 30 seconds to add up his modifiers FOR EACH ROLL (despite being written on the character sheet) - seems like an unachievable Holy Grail. It seems like you need to focus on just one aspect ("Skill" or "Weapon") to keep the streamline. the Greataxe outperforms the dagger in a "you only hit if you do more damage than the armor" system; the dagger is just as useful as the greataxe in a "I have more skill" (or "I am luckier with dice!") system.
Every time I think I have a system that does both, my players inform me "this is better run by a computer, to do all the math and fiddliness." (Pillars of Eternity comes to mind, with weapons modifying attack speed, bonus/penalty against certain armors, different skills modifying different weapon and armor effects... and in the end, I took the weapon and armor that looked cooler in my head, and ignored the modifers.)
I love the distinction in the mecha game Heavy Gear between "talent" (a plus on the d6 roll) and "experience" (rolling multiple d6s, hoping one meets the Targe Number). The young gun prodigy with d6+2 makes that TN 4 shot most of the time, but sometimes misses; the veteran with 3d6 almost never misses - one of those d6s is always 4 or higher - but that TN 7 shot requires 3 "6s" to pull off while the prodigy hits it 1in3. (yeah, sorry, that's a tangent again, because it's a hit roll, then damage, system.)
the way I see it the key thing about exploding dice as a mechanic is that it determines things by rolling a max die first. Without that the mechanic isn't one, so whether Im rerolling dice and/or keying other things off of it is immaterial imo. 
