This is just confusing, and confusing to avoid something that only has to look stupid if you want it to look stupid.
Not quite. The reason to do this isn't just to avoid showing Fingon hacking at a cliff with a knife/sword/whatever he has with him that isn't the harp or the bow. It's also to introduce Thorondor's involvement in a way that makes the reaction of the audience mirror the reaction of the elves.
The Hosts were right to pan some of the possible ways to introduce Thorondor here. Fingon should be surprised that his prayer is answered, which is fine, but what kind of startling are we going for? As Ange1e4e5 points out...not a jump scare. And not a slow-mo eagle flying in. So...he just shows up? And Fingon, what, asks for a ride? Handling the 'S'up, Fingon' appearance of Thorondor is something that has to happen if we're going to show that scene all the way through Fingon getting up on the cliff.
Not that it can't be done....but Thorondor flying into the camp of the Noldor has a much greater chance of success as an introduction. That one can work without question. Now...if we do want to show Thorondor's arrival in Thangorodrim, we have a lot of choices to make about how to pull that off. Possibly, Thorondor could swoop in, pick up Fingon, and deposit him on the ledge, all without conversation. Also possibly, Fingon and Thorondor could have a conversation about what type of assistance Fingon needs. But....the eagle is huge. It's going to get really crowded on that ledge, and we're going to wonder why the eagle's beak/talons couldn't cut through that iron band holding Maedhros in place. It's just...really difficult to ignore a giant eagle once you have one, and if the scene is meant to be about Fingon....
Not that I don't see your point about the climax being something we can't skip over. I think it's very clear that we intend Fingon to be our point-of-view character in this episode, and the struggle he's facing is making things right between the two camps personified in his efforts to rescue Maedhros. And so...we can show defeat after defeat as he keeps trying, but if we seemingly end on a note of defeat....how then is Maedhros rescued? Making the arrival back at the camps the climax (of both Fingon's story and the subplot involving Fingolfin) is probably not the easiest thing to do, but wasn't an awful idea, either.
If we plan to skip such a big scene as Maedhros' hand getting chopped off, then we do have to think carefully about what hints we've given the audience and how we convey to them what happened.
Idea that literally just came to me as I was typing above, and therefore isn't fully thought out yet... could we move the location of the eucatastrophic arrival? Have Fingon find his friend and climb up, try to free him, and then be prepared to kill him since there's no way to free him except cut off his hand, and there's no way to climb down, even for a Calaquendi, without two strong hands? That might remove the Hosts' concern that the eucatastrophe gets interrupted by futile, possibly comic, actions. But it could open up a whole new host of problems...
Yeah, I keep coming back to this. In the book, Thorondor helps with two things - lifting Fingon up the mountain so he can reach Maedhros on the cliff, and also carrying them both back to Mithrim. If we didn't want to show the Eagle hanging around while Fingon tries to cut Maedhros free, the solution is to have Fingon climb the cliff on his own (not actually an impossible task).
But...then when does he make the prayer to Manwë? And why does Thorondor arrive? Not that the taxi service isn't helpful, but....if that's all he's doing, it's potentially almost an afterthought to have him there. And if he arrives while Fingon is trying to cut through the iron cuff, we would expect the eagle to cut/bite through that for him. When the eagle just stands by and watches Fingon cut Maedhros' hand off....that might look...superfluous?