I understand this issue has been discussed before, and that this seems to be a known bug/issue with gamma and QuickTime. I did a fair amount of research, and it seems like the only fix was to use QuickTime pro and change a setting in "show movie properties". Most of the posts regarding this are from 5+ years ago, so I wanted to see if there was another way to fix this issue, especially as I don't have QT pro. I have tried rendering and exporting with a few different variations including animation, none, and h.264, but nothing seems to do the trick. I am also seeing this washed out/brightened effect while working in FCP.
I am using FCP6, and Adobe After Effects CS6 (I know, outdated)
For H.264 deliverables, try using Media Encoder to compress H.264/MP4 files instead of MOV files.
For Animation codec, FCP may incorrectly guess gamma for RGB-encoded video (like the Animation codec). Import the clip into FCP. Select the clip, then hit Cmd-9 to open Item Properties. Right-click the gamma value (normally set to Source) and set it to 2.2.