It seems to me, based on the Silmarillion’s account of the rescue of Maitimo from Angband, that Manwë didn’t send an Eagle specifically to aid the rescue, but that the Eagles were already present. Findekáno also cried out for help right before shooting, which is why an Eagle turned up then.
This makes a lot more sense than what I’d previously understood of the glowfic account:
- No echo of their lamentation needed to pass over the mountain; help was already there
- You don’t need a near-infinite air speed to get there from Valinor if you’re already present
- Why then? Well, because that’s when he asked for help. And, since he was sneaking around in magic darkness, that might also be the first time the Eagle saw him, although any explanation that relies on eagles of any capitalization not seeing something is suspect.
Manwë or Thorondor could definitely still have been more helpful, but it wasn’t quite as bad as I’d thought.