How to implement my own icons instead of using the svg icon system

You could replace the function that is getting the svg image like so:

<nav class="social-navigation" role="navigation" aria-label="<?php esc_attr_e( 'Footer Social Links Menu', 'twentyseventeen' ); ?>">
                    <?php
                        wp_nav_menu( array(
                            'theme_location' => 'social',
                            'menu_class'     => 'social-links-menu',
                            'depth'          => 1,
                            'link_before'    => '<span class="screen-reader-text">',
                            'link_after'     => '</span><img src="'.get_home_url().'/path/to/image.png">',
                        ) );
                    ?>
                </nav><!-- .social-navigation -->