My earliest gaming experience with combat initiative in Dungeons and Dragons was of the semi-random variety. Both sides (the party and the DM) rolled a d6, and whichever side rolled higher had initiative that turn.
As we became "more sophisticated" gamers, we turned to individual initiative, with each player rolling a d10 and adding their Dexterity bonus. The DM would also roll a d10, and attack priority would be granted from highest to lowest number, with the monsters attacking at the same time, based on the DM's roll. At one point, we even applied weapon speeds to our initiative rolls, so that players with faster weapons had a better chance of striking first.
I find this table (from the Ready Ref Sheets) to be very curious. It suggests that, after "Glance", "Breath" and Missile weapons, the longer and slower weapons have priority during combat. To be sure, a character weilding a faster weapon can modify the timing of their attack by having high Dexterity and wearing light or no armor, but the table still suggests that on balance it is the longer, slower melee weapons that act first.
For me, the attack priorities proposed in this table only make sense during the first round of combat, when the combatants first come to blows. After that, i'm not sure I would use this table to determine attack priority. I might still give Glance, Breath and Missile weapons the advantage, but would then reverse the order of the melee weapons, so that the short weapons would have earlier attack priority.
In modern Dungeons and Dragons, i'm not sure it really matters who strikes first (particularly when your characters have advanced several levels). But in Chainmail's Man-To-Man Combat section and Fantasy Supplement, attack priority is of critical importance. That is because, before the innovation of Hit Points, a successful hit resulted in an automatic kill. Therefore, whoever gained attack priority possessed a huge advantage, since, if their attack was successful, they would kill their opponent, thereby avoiding being killed themselves.
This could explain why attack priority and initiative was taken very seriously in early versions of Dungeons and Dragons.