Ideas on how to recover data from a screenless/headless Android device?
do you have developer options enabled?

If yes try to backup using adb. you will not have to access Adb shell. just 'adb backup' command will do it. use appropriate options to backup your shared storage. (just type adb and hit enteryou will see the Adb backup options)

go to following link try the options in answer to access Adb backup. do not encrypt the backup.

If u do not have developer access, just replace the device screen, and u should get your device as is.