After setting password and add"--unrestricted" in every entry, both "choosing entry to boot" and "pressing e key to edit" don't need password.
--unrestricted is set on an entry, only booting is allowed without the need to enter username and password.
Editing menu entries (e) and the use of the command line (c) is restricted to the users listed in the superusers list.
From the GRUB manual (emphasis mine):
In order to enable authentication support, the ‘superusers’ environment variable must be set to a list of usernames,
separated by any of spaces, commas, semicolons, pipes, or ampersands. Superusers are permitted to use the GRUB command line, edit menu entries, and execute any menu entry.
If ‘superusers’ is set, then use of the command line and editing of menu entries are automatically restricted to superusers.
Setting ‘superusers’ to empty string effectively disables both access to CLI and editing of menu entries.
Other users may be allowed to execute specific menu entries by giving a list of usernames (as above) using the --users
option to the ‘menuentry’ command (see menuentry).
If the --unrestricted option is used for a menu entry, then that entry is unrestricted. If the --users option is not used for a menu entry, then that only superusers are able to use it.
To be able to boot all entries without password, set
set superusers=... (empty list or list of usernames plus one password entry for each user) and add the
--unrestricted option to all menu entries.