The_excerpt() doesn’t parse – how to change that?

I do not know why the WordPress powers that be decided to do it the way they did but they don’t consider the part before the <--more--> to be the same as the excerpt. Whether that makes sense or not is probably an irresolvable matter of perspective. I’d accept it and make do. The distinction actually adds flexibility to your content management so long as you are aware of what is happening. If you want to use the <--more--> just do this…

if ( !empty( $post->post_excerpt ) ) the_excerpt();
else the_content();

The code that parses the <--more--> is hard-coded into get_the_content so it isn’t just a matter of removing/adding a filter, though certainly you could juggle filters, or create one, to make this work. That is a more complicated solution and is in my opinion not worth it.