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<?php

namespace Vendor\MiniCart\Controller\Cart;

use Magento\Framework\Controller\Result\JsonFactory;
use Magento\Framework\Json\Helper\Data;
use Magento\Framework\App\Action\Action;
use Magento\Framework\App\Action\Context;
use Magento\Checkout\Model\Session;
use Psr\Log\LoggerInterface;

class EmptyCart extends Action
{
    /**
     * @var Session
     */
    protected $checkoutSession;

    /**
     * @var JsonFactory
     */
    protected $jsonFactory;

    /**
     * @var Data
     */
    protected $jsonHelper;

    /**
     * @var LoggerInterface
     */
    protected $logger;

    /**
     * EmptyCart constructor.
     *
     * @param Context $context
     * @param Session $session
     * @param JsonFactory $jsonFactory
     * @param Data $jsonHelper
     * @param LoggerInterface $logger
     */
    public function __construct(
        Context $context,
        Session $session,
        JsonFactory $jsonFactory,
        Data $jsonHelper,
        LoggerInterface $logger
    ) {
        $this->checkoutSession = $session;
        $this->jsonFactory = $jsonFactory;
        $this->jsonHelper = $jsonHelper;
        $this->logger = $logger;
        parent::__construct($context);
    }

    /**
     * Ajax execute
     *
     */
    public function execute()
    {
        $response = [
            'errors' => false
        ];

        if ($this->getRequest()->isAjax()) {
            try {
                $this->checkoutSession->clearQuote()->clearStorage();
                $response['message'] = __('Empty Cart.');

            } catch (\Exception $e) {
                $response = [
                    'errors' => true,
                    'message' => __('Some thing went wrong.')
                ];
                $this->logger->critical($e);
            }
        } else {
            $response = [
                'errors' => true,
                'message' => __('Need to access via Ajax.')
            ];
        }
        /** @var \Magento\Framework\Controller\Result\Raw $resultRaw */
        $resultJson = $this->jsonFactory->create();
        return $resultJson->setData($response);
    }
}

<?php

namespace Vendor\MiniCart\Controller\Cart;

use Magento\Framework\Controller\Result\JsonFactory;
use Magento\Framework\Json\Helper\Data;
use Magento\Framework\App\Action\Action;
use Magento\Framework\App\Action\Context;
use Magento\Checkout\Model\Session;
use Psr\Log\LoggerInterface;

class EmptyCart extends Action
{
    /**
     * @var Session
     */
    protected $checkoutSession;

    /**
     * @var JsonFactory
     */
    protected $jsonFactory;

    /**
     * @var Data
     */
    protected $jsonHelper;

    /**
     * @var LoggerInterface
     */
    protected $logger;

    /**
     * @var Magento\Checkout\Model\Cart
     */
    protected $cart;

    /**
     * EmptyCart constructor.
     *
     * @param Context $context
     * @param Session $session
     * @param JsonFactory $jsonFactory
     * @param Data $jsonHelper
     * @param LoggerInterface $logger
     */
    public function __construct(
        Context $context,
        Session $session,
        JsonFactory $jsonFactory,
        Data $jsonHelper,
        LoggerInterface $logger,
        \Magento\Checkout\Model\Cart $cart
    ) {
        $this->checkoutSession = $session;
        $this->jsonFactory = $jsonFactory;
        $this->jsonHelper = $jsonHelper;
        $this->logger = $logger;
        $this->cart = $cart;
        parent::__construct($context);
    }

    /**
     * Ajax execute
     *
     */
    public function execute()
    {
        $response = [
            'errors' => false
        ];

        if ($this->getRequest()->isAjax()) {
            try {
                $this->cart->truncate()->save();
                $response['message'] = __('Empty Cart.');

            } catch (\Exception $e) {
                $response = [
                    'errors' => true,
                    'message' => __('Some thing went wrong.')
                ];
                $this->logger->critical($e);
            }
        } else {
            $response = [
                'errors' => true,
                'message' => __('Need to access via Ajax.')
            ];
        }
        /** @var \Magento\Framework\Controller\Result\Raw $resultRaw */
        $resultJson = $this->jsonFactory->create();
        return $resultJson->setData($response);
    }
}

define([
    'jquery',
    'Magento_Checkout/js/view/minicart',
    'Magento_Ui/js/modal/alert',
    'Magento_Ui/js/modal/confirm'
], function ($ ,Component, alert, confirm) {
    'use strict';

    return Component.extend({
        confirmMessage: $.mage.__('Are you sure you would like to remove all items from the shopping cart?'),
        emptyCartUrl: window.checkout.emptyMiniCart,

        emptyCartAction: function () {
            var self = this,
                href = self.emptyCartUrl;
            $('#empty-minicart').on('click', function () {
                var el = this;
                confirm({
                    content: self.confirmMessage,
                    actions: {
                        confirm: function () {
                            self._removeAllItems(href, el);
                        },
                        always: function (event) {
                            event.stopImmediatePropagation();
                        }
                    }
                });
            });
        },

        _removeAllItems: function (href, elem) {
            $.ajax({
                url: href,
                type: 'post',
                dataType: 'json',
                beforeSend: function () {
                    $(elem).attr('disabled', 'disabled');
                },
                complete: function () {
                    $(elem).attr('disabled', null);
                }

            }).done(function (response) {
                if (!response.errors) {

                } else {
                    var msg = response.message;

                    if (msg) {
                        alert({
                            content: msg
                        });
                    }
                }
            }).fail(function (error) {
                console.log(JSON.stringify(error));
            });
        }
    });
});

 <!-- ko if: getCartParam('summary_count') -->
    <div class="actions">
        <div class="secondary">
            <a class="action empty-cart" id="empty-minicart" data-bind="afterRender: emptyCartAction()">
                <span><!-- ko i18n: 'Empty Cart Now' --><!-- /ko --></span>
            </a>
        </div>
    </div>
 <!-- /ko -->

   define([
    'jquery',
    'Magento_Checkout/js/view/minicart',
    'Magento_Ui/js/modal/alert',
    'Magento_Ui/js/modal/confirm'
], function ($ ,Component, alert, confirm) {
    'use strict';

    return Component.extend({
        confirmMessage: $.mage.__('Are you sure you would like to remove all items from the shopping cart?'),
        emptyCartUrl: window.checkout.emptyMiniCart,

        emptyCartAction: function (element) {
            var self = this,
                href = self.emptyCartUrl;
            $(element).on('click', function () {
                var el = this;
                confirm({
                    content: self.confirmMessage,
                    actions: {
                        confirm: function () {
                            self._removeAllItems(href, el);
                        },
                        always: function (event) {
                            event.stopImmediatePropagation();
                        }
                    }
                });
            });
        },

        _removeAllItems: function (href, elem) {
            $.ajax({
                url: href,
                type: 'post',
                dataType: 'json',
                beforeSend: function () {
                    $(elem).attr('disabled', 'disabled');
                },
                complete: function () {
                    $(elem).attr('disabled', null);
                }

            }).done(function (response) {
                if (!response.errors) {

                } else {
                    var msg = response.message;

                    if (msg) {
                        alert({
                            content: msg
                        });
                    }
                }
            }).fail(function (error) {
                console.log(JSON.stringify(error));
            });
        }
    });
});

 <!-- ko if: getCartParam('summary_count') -->
    <div class="actions">
        <div class="secondary">
            <a class="action empty-cart" id="empty-minicart" data-bind="afterRender: emptyCartAction">
                <span><!-- ko i18n: 'Empty Cart Now' --><!-- /ko --></span>
            </a>
        </div>
    </div>
 <!-- /ko -->

[IMAGES DEMO]

Mini Cart:

Message Confirmation: