Issue displaying customiser setting with get_theme_mod

The settings attribute you’re using for add_control() is expecting an array of all settings tied to the control while you’re passing a string. The attribute you want is the singular setting which is expecting the string value of the primary setting for the control.

So try this:

$wp_customize->add_control( 'button_1_text', array(
    'label'    => __( 'Primary Button', 'test_theme' ),
    'section'  => 'header_buttons',
    'setting' => 'button_1_text_settings',
    'type'     => 'text'
));

It’s also not clear to me if it’s best practice, but I tend to use the same string as my setting ID for my control ID, then I can just omit the setting attribute since it defaults to the ID, like this:

$wp_customize->add_setting( 'button_1_text', array(
    'type' => 'theme_mod',
    'default' => __('Primary Button Text', 'test_theme'),
    'transport' => 'postMessage'
));
$wp_customize->add_control( 'button_1_text', array(
    'label'    => __( 'Primary Button', 'test_theme' ),
    'section'  => 'header_buttons',
    'type'     => 'text'
));