set_query_var doesn't seem to work on init hook

Question

Based on the trick of using set_query_var with get_template_part to easily pass variables from one template file to another (see here: Method 3), I thought I could use the same method for passing complex information back to a page after a form submission (error messages, next steps, topic URLs, etc.)

Very simply, I have an init hook listening for the form submission set up like this:

function vnmProcessForm() {
    if (isset($_POST['form-value'])) {

        $formVal = sanitize_title($_POST['form-value']);

        if ($formVal == 'all-done') {
            set_query_var('form-message', 'It looks like you are all done with this step! Now move on to <a href="/dashboard">setting up your dashboard</a>.');
        } else {
            set_query_var('form-message', 'Please complete all options in the form.');
        }

        // It's a lot more complex than this, but... MVC.
    }
}

add_action('init', 'vnmProcessForm');

My form action is the same URL that the form resides on (so, essentially, a refresh); however, when I try get_query_var('form-message') on that page, I get nothing. If I spit out all query_vars, form-message isn't among them.

I'm sort of perplexed as to why this would work seamlessly for something like get_template_part, yet fails to work in this instance. Is it something to do with the init hook being too early?

In the meantime I have 'solved' this by simply using $_POST instead of set_query_var, but I'm genuinely curious to know why that isn't working.