After taking some additional time to ruminate on my three requirements for an initiative system, I decided to take a stab at making an initiative system specifically with those three requirements in mind. Something which maximized excitement, without requiring too much time or attention from the players.
Players roll initiative individually, based on the speed of the weapon they are using. Most weapons, and any character who is not using weapons, roll 1d6 for their initiative. Characters using slow weapons, such as the sword pictured, roll 1d4 for their initiative. While characters using fast weapons, such as rapiers, loaded loaded crossbows, or daggers, roll 1d8 for their initiative. Slow, Average, and Fast weapons align nicely with the great, medium, and small/minor weapons in Lamentations of the Flame Princess.
There are no modifiers to these rolls. However, characters roll their initiative die a number of times equal to their dexterity bonus or penalty (with a minimum of 1 die rolled). If the character has a bonus, they take the best result of the dice they rolled, and drop the rest. If the character has a dexterity penalty, they do the opposite. Using only the worst result.
So a fellow with a -2 dexterity modifier wielding a Zweihänder would roll 2d4, and whichever result was lower would be their initiative. Meanwhile, a fellow with +2 dexterity wielding a dagger would roll 2d8 and take whichever result was higher as their initiative. Characters with initiatives of 1, -1, and 0 would all roll only a single die.