Is there a way to get a path to the theme directory without the server name?

Yes get_template_directory:

get_template_directory(): string

Retrieves template directory path for the active theme.
Returns an absolute server path (eg: /home/user/public_html/wp-content/themes/my_theme), not a URI.

In the case a child theme is being used, the absolute path to the parent theme directory will be returned. Use get_stylesheet_directory() to get the absolute path to the child theme directory.

To retrieve the URI of the stylesheet directory use get_stylesheet_directory_uri() instead.

https://developer.wordpress.org/reference/functions/get_template_directory/

There’s also get_stylesheet_directory which refers to the current theme, whereas get_template_directory always refers to the parent theme ( if there is no child theme then they will return the same value )

$svg = file_get_contents( get_template_directory() . '/file.svg' );