Run ffmpeg to remove exif on several files at once
I ended up going with for f in /path/*.MTS; do
  ffmpeg -i "$f" -map_metadata -1 -c:v copy -c:a copy "$f copy.MTS" 
done
and came up with files like "video 1.MTS copy.MTS" which work just fine and with no timestamp.

There might be a more logical way to do it but as I'm no expert when it comes to shell script I'm gonna stick with this one in case I need it again.

Thanks to Graham Miln for helping, cheers everyone !