"~/Desktop/test.txt: No such file or directory"

Try replacing ~ with $HOME. Tilde expansion only happens when the tilde is unquoted. See info "(bash) Tilde Expansion".

You could also do file=~/Desktop without quoting it, but if you ever replace part of this with something with a field separator in it, then it will break. Quoting the values of variables is probably a good thing to get into the habit of anyway. Quoting variable file=~/"Desktop" will also work but I think that is rather ugly.

Another reason to prefer $HOME, when possible: tilde expansion only happens at the beginnings of words. So command --option=~/foo will only work if command does tilde expansion itself, which will vary by command, while command --option="$HOME/foo" will always work.

FYI, you can also use eval:

eval "echo "TESTING" > $file"

The eval takes the command as an argument and it causes the shell to do the Tilde expansion.