Why doesn’t my site autosave?

You are looking for the reference of auto-save in wordpress.com. I think you can have a look to your wp-config.php to see if you have a define about the auto-save interval.

When editing a post, WordPress uses Ajax to auto-save revisions to the post as you edit. You may want to increase this setting for longer delays in between auto-saves, or decrease the setting to make sure you never lose changes. The default is 60 seconds.

Editing wp-config.php

define( 'AUTOSAVE_INTERVAL', 160 ); // Seconds

If you find this line, you can remove it, and the autosave functionnalities will come back. If you use any cache plugin, you maybe need to manage its options.