I don’t know about eagles, but I’ve never seen an osprey pick up a floating fish. I’ve seen them turn, swoop down to take a better look and then go on their way, but never pick one up, even if flopping.
This was observed over many years on both salt and fresh water.
Eagles will pick up floaters. When fishing on a lake we would sometimes catch fish that could not be released with any chance of survival. Nearby eagles would immediately swoop in and take the fish back to the nest.
On the river we fish on when we are near the eagles nest we will throw out any dead fish we have up stream from the nest… the eagle will launch from the nest and have a look at the morsel and then make a run on it with its talons outstretched and snatch the fish and return to its nest. Quite awesome to watch and record a video of the action.