Why doesn't composer install drush on $HOME/bin directory?
Composer downloads things and puts them where you (read: your composer/project settings) tells them to go; or in the default location if you haven't specified. It won't assume you want to link anything from the downloaded library into /usr/bin, though this could theoretically be done through a post-install-cmd in your composer.json file.

If you install something globally you'll find it in ~/.composer/vendor/ (the Drush executable will probably be in ~/.composer/vendor/bin/drush).

You've probably missed this important step from the instructions to install Drush via Composer:


  Make sure Composer's global bin directory is on the system PATH


Once Composer's global bin path is in your system path, the drush command will start working immediately.