Material tab bar with icons

custom.css

.site__navigation .tab-bar .mdc-tab .material-icons {
    color: var(--mdc-theme-on-header, var(--mdc-theme-on-primary, #fff));
}

custom_walker.php

<?php

namespace child_theme\material;

/**
 * Copied from MaterialDesign\Theme\Menu_Walker.
 *
 * Additional changes are in span.mdc-tab__content.
 */
class Custom_Walker extends Walker {
	/**
	 * DB fields to use.
	 *
	 * @var array
	 */
	public $db_fields = [
		'parent' => 'menu_item_parent',
		'id'     => 'db_id',
	];

	/**
	 * Add necessary classes to menu items
	 *
	 * @param string $output Used to append additional content (passed by reference).
	 * @param object $item   The data object.
	 * @param int    $depth  Depth of the item.
	 * @param array  $args   An array of additional arguments.
	 * @param int    $id     ID of the current item.
	 */
	public function start_el( &$output, $item, $depth = 0, $args = [], $id = 0 ) {
		$output .= sprintf(
			'<a href="%1$s" %2$s>%3$s</a>',
			esc_url( $item->url ),
			( in_array( 'current-menu-item', $item->classes, true ) ) ? ' class="mdc-tab mdc-tab--active" aria-selected="true"' : ' class="mdc-tab"',
			$this->build_markup( $item )
		);
	}

	/**
	 * Add necessary markup to build tab
	 *
	 * @param  object $item The data object.
	 * @return string Markup to display
	 */
	private function build_markup( $item ) {
		$is_active = in_array( 'current-menu-item', $item->classes, true );

		ob_start();
		?>
		<span class="mdc-tab__content">
			<?php
			echo wp_kses(
				$item->title,
				[
					'span' => [
						'class' => true,
						'aria-hidden' => true,
					],
				]
			);
			?>
		</span>

		<span
			class="mdc-tab-indicator
			<?php
			if ( $is_active ) {
				echo 'mdc-tab-indicator--active';
			}
			?>
		">
			<span class="mdc-tab-indicator__content mdc-tab-indicator__content--underline"></span>
		</span>
		<span class="mdc-tab__ripple"></span>
		<?php

		return ob_get_clean();
	}
}

functions.php

<?php 

namespace child_theme\material; 

add_filter( 'wp_nav_menu_args', __NAMESPACE__ . '\\override_material_walker' );

function override_material_walker( $args ) {
	require_once get_template_directory() . '/inc/custom_walker.php';
	$args['walker'] = new Custom_Walker();

	return $args;
}

menu-item-sample.html

<span class="mdc-tab__icon material-icons" aria-hidden="true">home</span><span class="mdc-tab__text-label">Home</span>