It's kind of a standard thing in 3.x, though; if you jump, and just miss (fail by less than 5), you get a Reflex save to grab hold of the far edge.
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He missed by one; a save to avoid going ker-splat-sizzle seems fair. Denying it would seem kind of unfair, to me.
I admit I'd be likely to give him the "second chance" as well, but that's largely because he rolled a 12 when he needed a 13. I'm superstitious enough to say that's too close not to be taken into account. The ruling of making a saving throw jibes with the overall rule of getting a save to prevent you from plummeting if you're sufficiently near a handhold, and a 12 says "you are really near that handhold."
Pretty much my take as well. If he'd needed 13 and rolled a 5, he's missed by miles and is well on his way to lava-induced immolation but such a close roll to me is "missed it by that much" territory and, given the self-preservation instinct, I'd say the character would automatically attempt to grab it without even consciously thinking of it. A "roll to see if your frantic snatching saves you" for near misses that could be reasonably viewed to be within arm's reach is fine by me.I actually agree that if the GM had allowed unlimited saving throws that would suck BUT in this case the player only failed by one. That to me means his PC just missed the wyvern by a whisker and is flying past. Totally reasonable to allow another save to grab it on the way down imho.
If that had failed and it was followed up with "OK, roll to see if a thermal updraft from the hot lava pushes your character upwards.... No? OK, roll to see if an air elemental notices your plight..." then I'd have serious issues.
So far as I could tell when I read the OP, the situation and its resolution were fair. A near miss like that practically demands a "saving throw" - else "fridge logic" would eventually cut in and everyone - players and GM - would be saying later "yeah, but if you were falling down a cliff you'd instinctively make a frantic grab for anything to break your fall" - and by then it'd be too late to "rewrite" the character's survival.
A substantial miss, however and the only thing you can do is start making comments on how the character chose an inopportune moment to spectacularly commit suicide - "in the middle of a pitched battle 'n' all, just when we could have done with his help"...
As to the "cool" factor:
I never once played a game that wouldn't be "cooler" than real life - grizzled space mercenaries and adventurer/traders in Traveller, various fighters, magic users, beast masters (including a female Drow and a Wemic) in AD&D, gritty edgerunners in Cyberpunk, Vampires and Werewolves in Monster: The Something, Goth paranormal investigator in nWoD - and that's getting close to thirty years of gaming...