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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' ) ) {
}