Developer who added custom post types at your site registered theme with user capabilities default for the WordPress built-in posts, but protected admin menu items by ‘edit_pages’ capability.
I used “Admin menu access” add-on to get this information: https://www.role-editor.com/block-admin-menu-items
Then I added ‘edit_pages’ capability to the ‘Speaker’ role and blocked unneded menu items with the same “Admin menu” button.
Tests showed that while custom post types are defined with default ‘post’ capability type, they require really capabilities from the ‘pages’ group. So I added them to the ‘Speaker” role.