I got this triumph a week or two ago, so I can't speak to feathers this week. One thing that I thought I noticed but I'm not 100% sure of is that the feathers might visually disappear if someone else on your team claims them, but if you go to the exact right spot, you can still press 'e' to pick them up.
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Good idea but sadly didn't work in my case. On the first run I was the first one through the portal and at the feather location, no feather and no pick up prompt. Second time around I made sure to be the last one through the portal as I thought maybe it's a loading/timing issue with the fireteam but still with the same result.
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Sounds like you should just try it a different week when the locations change. Seems like this layout may be bugged.