This seems related to #718397, but not exactly the same.

Running gnome-terminal on a new user account I cannot make it remember
the fact that I unchecked the "Show Menu bar" option.  When I
close my current terminal window and open a new one, the menu bar is
there again.

When I check gsettings, it is always set to true.

$ gsettings get org.gnome.Terminal.Legacy.Settings default-show-menubar
true

Setting it manually to false is a workaround, but there is something
broken in the UI that prevents it from saving the setting properly.

Hi,

Right click menu's "Show Menubar" by design only influences the
current window (similarly to some other options there, such as
selecting a Profile or toggling Read-Only).

What you're looking for is the global menu's Edit -> Preferences ->
"Show menubar by default in new terminals".

cheers,
egmont