Then why doesn't someone other than Sadeas try smaller bridges that armored and armed men carry to the chasms? A ten-man team could carry a 30' walkway-type bridge while a few other warriors hold shields to deflect arrows. Lay the bridge and the small group of men cross to create a staging area for more bridges and men. Slower, larger bridges arrive later. The smaller bridges could even interlock to make a larger bridge for more men. That would be faster than a large bridge.
I don't know. I guess I'm just having trouble comprehending how the bridges and bridgemen survive even ONE assault when in every other piece of fantasy I've ever read and assault like that would fail spectacularly. As I said before, this is the one thing about the book that is causing me to pull my hair out in frustration.
Many bridge crews
don't survive, a good concentrated volley of arrows is sufficient to wipe out a crew (or at least stop them from placing their bridge). This is why Sadeas attacks with as many as 20 bridges in a single assault, and while Brandon doesn't call out numbers I would guess that maybe half those bridges are actually laid (he only says that enough bridges are laid to effect a cavalry charge).
You're right in that the tactics you describe would be more effective in saving lives, but as Kaladin comes to realize,
bridgemen are supposed to die. They're moving targets, cannon-fodder who's apparent purpose is to lay bridges, but their actual purpose is to draw fire from the fighting troops, because the Parshendi don't appear to very discriminating in terms of target acquisition. I'm surprised that Sadeas doesn't make all his bridgemen wear giant bulls-eye targets on their tunics.
Parshendi appear to run out of arrows before too long, this is noted in a couple engagements (most notably Kaladin's final). And it really isn't clear how well-supplied either army actually is with arrows... archery isn't mentioned a lot, wood is in short supply, and I don't even know what they use for fletching (no birds on Roshar, remember?). So while they do fire arrows at each other in the initial stages of each battle, that seems to end pretty quickly and it gets down to the choppy work.
On the topic of "fire arrows", I'd point out that flaming arrows are very cool in Hollywood, but less so in reality.
Adding an additional weight (usually a tow made of pitch-soaked string or cloth which is then set on fire) to an arrowhead is a certain way to screw up its ability to fly true, because you've just messed up the balance. There are cases of them being used in warfare, but in all the examples I've heard (with the possible exception of naval warfare) it's usually employed for "plunging fire", where the archers release at a high angle and precise aim is not the point, but rather blanket coverage in the hopes of starting a blaze.
The exchanges of arrows in Plains battles are, I think, almost always "directed fire", with archers firing in low arcs directly at the enemy. While it's sort of easy to hit targets that are bunched together, I'm sure there's plenty of misses too (more so if they're shooting back), and I don't see the need for either army to waste a limited supply of ammunition on a tactic of questionable benefit. As Kaladin learns, in the Alethi army innovation is not rewarded. And it doesn't seem to be part of the Parshendi strategy at all.