Setting a custom sub-path for blog without using pages?

^{I don’t have time to explain this in detail (I shall upon return) but in the meantime this should work for you,}

^{Answer updated with explanation as promised.
WP Rewrite rules are like voodoo, I'm sure there's more than one way to go about this, but here's mine.}

Problem:

Just to clarify your question for others who may stumble upon this thread, what you want to do is create a page without having to physically create an empty-placeholder-page within the administration dashboard found under: Page -> Add New.

Essentially, you want to create a Fake Page and have that page use any template you specify.

Solution:

First we setup our rewrite rules,

add_action('init', 'fake_page_rewrite');

function fake_page_rewrite(){

    global $wp_rewrite;

    //set up our query variable %fake_page% which equates to index.php?fake_page= 
    add_rewrite_tag( '%fake_page%', '([^&]+)'); 

    //add rewrite rule that matches /blog/page/2, /blog/page/3, /blog/page/4, etc..
    add_rewrite_rule('^blog/page/?([0-9])?','index.php?fake_page=blog&paged=$matches[1]','top');  

    //add rewrite rule that matches /blog
    add_rewrite_rule('^blog/?','index.php?fake_page=blog','top');

    //add endpoint, in this case 'blog' to satisfy our rewrite rule /blog, /blog/page/ etc..
    add_rewrite_endpoint( 'blog', EP_PERMALINK | EP_PAGES );

    //flush rules to get this to work properly
    $wp_rewrite->flush_rules();

}

Within the add_rewrite_tag we specify our query variable in the form of %fake_page%, which by the way you can specify whatever you want or appropriate for your needs. My example fake_page is only symbolic to illustrate the mechanics of this answer.

How the query variable works in this instance is by matching a request for,

http://www.example.com/blog

…which is then internally mapped to,

http://www.example.com/index.php?fake_page=blog

The latter being what you would see when running the default permalink structure.

In a similar fashion requests for,

http://www.example.com/blog/page/2
http://www.example.com/blog/page/3
http://www.example.com/blog/page/4
etc...

…would each map to their equivalents,

http://www.example.com/index.php?fake_page=blog&paged=2
http://www.example.com/index.php?fake_page=blog&paged=3
http://www.example.com/index.php?fake_page=blog&paged=4
etc...

In the example snippet above you will notice that we have the rewrite rule which firsts matches the pagination /blog/page/{page_number}, above our second rule which matches the base for our fake page of /blog.

This is necessary so that the base rule doesn’t return a match on the first occurrence of our endpoint which is defined as blog before having a chance to evaluate the rest of the requested URL to ensure that the user hadn’t in fact requested a paged result. Basically, reverse those rules and it doesn’t work.

As mentioned before, rewrite rules are like voodoo to me so there’s
probably another way to go about the order in which you specify
your rules which is possibly related to using the function add_permastruct . If
anyone has an alternative, then chime in!

The next function which hooks onto template_redirect checks for the existence of our fake_page within the query variables, if its matched/exists within the array we then request the inclusion of our desired template file to handle the presentation of data.

add_action('template_redirect', 'fake_page_redirect');

    function fake_page_redirect(){

        global $wp;

        //retrieve the query vars and store as variable $template 
        $template = $wp->query_vars;

        //pass the $template variable into the conditional statement and
        //check if the key 'fake_page' is one of the query_vars held in the $template array
        //and that 'blog' is equal to the value of the key which is set
        if ( array_key_exists( 'fake_page', $template ) && 'blog' == $template['fake_page'] ) {

            //if the key 'fake_page' exists and 'blog' matches the value of that key
            //then return the template specified below to handle presentation
            include( get_template_directory().'/your-template-name-here.php' );

            exit;

        }
    }

_{PS. I have tested this and it works under my conditions, though I'm not sure if any other quirks may popup with rewrite rules and end point masks the deeper you go, therefore you should thoroughly test all pagination and permalinks and make sure that they resolve to their intended paths correctly . If not, we can address those issues as they arise.}