get_theme_mod() has a second parameter where you can set the default value, the value to use if the option is not set.
So, if the default value is "1", you can use:
get_theme_mod( 'display_header_image_title', '1' )
Then, if there is not value for display_header_image_title (no value in database), "1" is used. So, you could check the exact value:
if( get_theme_mod( 'display_header_image_title', '1' ) === '1' ) {
}
Or just true/false:
if( get_theme_mod( 'display_header_image_title', '1' ) ) {
}