If you don't want single user mode to prompt for a password, the file responsible is /lib/systemd/system/rescue.service, specifically the ExecStart line, however, it's inadvisable to edit this file directly as it may get overwritten.
Instead, we need to create an override file for it like this:
systemctl edit rescue.service
This will open a text editor. Add these lines:
[Service]
ExecStart=
ExecStart=-/bin/bash
(FYI: the blank ExecStart= is mandatory to unset the existing value before setting it to what we want)
Now save the file, and it will create a file /etc/systemd/system/rescue.service.d/override.conf the contents of which will override /lib/systemd/system/rescue.service
You will also need to make sure the package friendly-recovery is uninstalled as it will prevent this from working
apt purge friendly-receovery
Uninstalling the package should automatically invoke update-grub but if it doesn't you may need to run it manually.
If you want to revert to the default behavior (prompt for root password), delete the /etc/systemd/system/rescue.service.d/override.conf file you created and (if desired) reinstall the friendly-recovery package.