I can't remember exactly how I ran it when I was GMing, but I think we went reach then speed. Reach gave you an opportunity attack on the way in, once past that it was dex + roll, modified by speed. Most of the high dex types went for fast weapons so they could get in a first strike, the lower dex ones went for slow but heavy weapons, 'cause if they survived the initial attack, they were going to deal a heap of damage.
We even had characters with long weapons (spears and polearms) behind the sword and shield fighting types, so they could take defensive advantage of the meat wall, oops, shield wall, yeah, that's what I meant, right, where was I? oh, yeah, in front of them.
Speed factors made sense. We had each weapon tagged with two numbers: reach, and initiative penalty (speed factor). It meant that people went for thrusting weapons or short weapons, so they could work together. It got intra-party coordination working well. Losing it, well, lost a lot of the flavor of the battles.

Initiative is an important aspect of an encounter. Who goes first often determines the winner or at least will help to minimize the damage sustained by the adventurers. I use casting time and speed factor for several reasons. The most important to me was the flavor of spell and weapon choice. Secondly it gave the heroes an advantage. With style specialization and reaction adjustments for high dexterity it was not difficult to ensure at least one player would get to act before the enemy.

It helps differentiate between fighter types and balances them. If you have a player that wants to play a tank, let them. If the character is decked out in full plate armor hefting a two handed sword be prepared to get into the fight last. If someone wishes to play the lightly armored swashbuckler type with the stinging rapier they’re going to benefit from the faster weapon. Now bring in weapon styles to really customize a character.

As for slowing down PBP, combat always slows down the game. There are many options to help that out. First try having the players post two sets of combat actions at a time. Consider using a minion rule. If the creature isn’t a boss, use one hit equals one kill. This is particularly appropriate when facing the horde of kobolds, orcs or whatever. Minimize combat time and get back to the story.

Weapon Speed Factor always had a place at every gaming table I ever sat at, and it just became second nature in all games. It got to the point where our group knew every weapon speed for every weapon without having to look it up. And we enjoyed using it.

One question I have for those people who don't use weapon speed: What did you do to make it balanced for spell casters? Say you have a fighter and a mage, enemies, who have rolled initiative. They both roll a d10 and they each get a 5. Neither has a dexterity bonus (for the purposes of this example). So, the mage starts casting magic missile on 5, and it has a casting time of 1 segment, so it won't go off until 6. But if the fighter doesn't have to use weapon speed, then his sword just strikes the mage on 5, which puts the mage at a serious disadvantage. It isn't balanced to take away weapon speed from fighters and rogues, but leave the casting time for mages and priests.

I've never played 2e, but I've always thought that the lack of a "speed" rating in most crunch-heavy RPGs (D&D 3.5 falls under this heading) was a bit of an oversight. Without speed as a factor, you lose a third dimension in dynamic combat. Now the spiked-chain crusader has the exact same reaction time as the dual-dagger rogue and the volley-archery ranger. Even an attempt to create the semblance of a speed difference by granting more attacks per round to the "faster" and "less-encumbered" characters would have gone a long way toward fixing that issue; instead, they locked the number of attacks per round into the class structure and feats.

Which I think is ultimately what made 2-handers so much more powerful than any other weapon group (putting aside the sheer disparity within the 2-hander group itself). At the end of the day, when you're getting the same number of attacks (or more in some cases) with a huge, clumsy weapon as you are with small, light, precise weapons, the bigger numbers are going to win. And from there, feats can go into making those numbers bigger, while the "fast" characters have to waste their feats on getting what should have been part of the basic character structure to begin with: additional attacks.

Of course, I think that "more attacks" is a very simplistic way of attributing a higher speed to a character (there's movement per round, increased AC, more actions per round, etc etc), but it's pretty much the only connection to the idea that D&D 3.5 has managed to work into its own system.