Spawn a terminal and redirect its stdout to original process

I'm trying to spawn a new terminal, execute a few commands and pipe their output to stdin of the original process. A mwe of what I'm trying to do is the following bash one-liner:

$ xterm -e sh -c "echo -e 'foo\nbar' > /proc/$$/fd/1" | grep foo

where I spawn a new xterm window, print foo\nbar and try to redirect it to stdin of the shell I'm executing this command from. To see if it works I then pipe it into grep. The code above prints both foo and bar to stdout instead of just foo, but I'm not sure why. How can I fix this? EDIT: the real case implementation of this is a file picker that uses fzf:

#!/usr/bin/env bash

function open() {
  # takes a bunch of file names (either passed as arguments or from stdin) and
  # opens them
  true
}

# This is what I have now
alacritty \
   -e sh -c 'fzf -m --prompt="Open> " --border=horizontal --print0 \
              | (nohup xargs -0 bash -c '\''open "$@"'\'' _ &>/dev/null &)'

# This is what I'm trying to do
# alacritty \
#  -e sh -c "fzf -m --prompt='Open> ' --border=horizontal > /proc/$$/fd/1" \
#  | open