Why redirection with tee doesn't work as expected?

Question

What I want to obtain is logging all the output of a script to a file, all in the script itself, saving all I see on the screen in the log file. If I don't redirect to file the script works as intended, continuing to loop forever till I hit CTRL+C; when this happens it prints a couple of messages on screen and quits. If I try to redirect to a file, using tee, not only it doesn't print the messages but also the log is incomplete, there's only the loop lines, nothing afte that. Here's the script (cut down version):

#!/bin/sh
EXIT=0
COUNT=0
{
while [ $EXIT -eq 0 ]
do
  COUNT=$((COUNT+1)) && echo "Eternal loop $COUNT" && sleep 1
  trap 'EXIT=1; echo "Loop interrupted by user, exiting."' 15 2 8 20
done
echo "Out of loop, eternity ends"
} | tee log.log 2>&1

I read around and couldn't understand if the issue I have is caused by piping to tee, by stderr and stdout, or by how I useda trap. I'm a noob at shell scripting and this is frustrating, any help will be appreciated. Thanks.

Edit 1: After reading chepner's answer I modified the script like this:

#!/bin/sh
EXIT=0
COUNT=0
rm log.log
while [ $EXIT -eq 0 ]
do
COUNT=$((COUNT+1)) && echo "Eternal loop $COUNT" | tee -a log.log && sleep 1
trap 'EXIT=1; echo "Loop interrupted by user, exiting." | tee -a log.log' 15 2 8 20
done
echo "Out of loop, eternity ends" | tee -a log.log

and it works, it logs the output to a logfile, and I still see the output on screen, exactly how I wanted. Will have to try it on the other script, but I suppose it'll be OK. Now a question remains: if I have lots of commands instead of just 3, would this solution be "clean" enough? I mean, what happens to all those tee I call forth? Is there a better way to do this, like more polished?

Answer 1

Not exactly what you're asking, but when you want to log everything you see during a process, the script command is easier than dealing with redirection.

if you want to use tee anyway, move the redirection in front of the pipe

somecommand 2>&1 | tee outfile