There is a way to make screen work in the way you describe. This is an issue with the dropbear configuration. By default, the SSH daemon (dropbear) on the BeagleBones are configured to kill all user procesess as soon as the user logs off. This makes it difficult to leave processes running in the background for debugging. To change this, edit the dropbear@.service file in /lib/systemd and add the "KillMode=process" line:
root@beaglebone:/lib/systemd/system# cat dropbear\@.service
[Unit]
Description=SSH Per-Connection Server
Requires=dropbearkey.service
After=syslog.target dropbearkey.service
[Service]
ExecStart=-/usr/sbin/dropbear -i -r /etc/dropbear/dropbear_rsa_host_key -p 22
ExecReload=/bin/kill -HUP $MAINPID
StandardInput=socket
KillMode=process
You can either reboot the system for these changes to take effect, or reload the systemctl config and restart the dropbear service.