The only drawback to your idea of sending Legolas after Frodo would be that the presence of an Elf would have raised greater interest from the Orcs at Cirith Ungol (who captured Frodo) than a couple of Halflings -- one of whom (Sam) they thought got carried off by Shelob. Neither the Orcs (nor Sauron, apparently) showed intense interest in Frodo once it had been determined that the ring was not with him. But an Elf with them as well? I'd suspect closer attention would have been paid in that event. Sending an Elf with the halflings would have raised their profile somewhat, and that could be disastrous.
And as it turns out, Legolas was pretty helpful at Helm's Deep.
Actually, my thought was that Legolas would have killed Shelob at Cirith Ungol. I realize that we are violating the "don't get ahead of the story" rule, but my thought was that one serious fighter might have been of value to Sam and Frodo. The fight at Cirith Ungol would have been the place where that value would be seen. I've forgotten the exact sequence there, but I remember a sense of panic that led to the hobbits being particularly vulnerable to the onset of the attack. I think Legolas would have helped them keep their composure and maybe even had the light out before Shelob made her move. I don't think Shelob would have attacked two elvin swords and the light. If she had, I think Legolas would have been able to kill her.
Obviously, Legolas couldn't have made the march passing for little orcs as Frodo and Sam did. I'm not criticizing the way that Tolkien wrote the story. No one can top the way Tolkien wrote the story. I'm just saying that had I been in Aragorn's position, I would have thought it out that way. I wouldn't have forseen the exact battle, but I might have sent him with that in mind.
WFTR
Bill