I guess this should have been obvious to me long ago, but if you have shell access you can near-instantly upgrade drupal like this:
wget (new drupal release)
tar xfz (new drupal release)
cd drupal-(new drupal version)
rm -rf sites
rsync .htaccess * (your drupal root)
If you need it spelled out a little more closely, here's what the commands look like for me for 7.18, which seems to be a security update...
wget http://ftp.drupal.org/files/projects/drupal-7.18.tar.gz
tar xfz drupal-7.18.tar.gz
cd drupal-7.18
rm -rf sites/
rsync -av .htaccess * ../public_html/
Before doing this log in as whoever runs update.php. After doing this, run update.php. Done and done. This assumes you haven't customized .htaccess. If you have, just don't mention .htaccess in your commandline. You might want to compare it to the .htaccess from the previous version to make sure you don't need to make changes.