Recently i have the same issues in one of my website, as i can see in your live website message is displaying but it clear immediately on load, when i inspect the message block in your site thare is no error message, you have to check following file
/vendor/magento/module-theme/view/frontend/web/js/view/messages.js
define([
'jquery',
'uiComponent',
'Magento_Customer/js/customer-data',
'underscore',
'jquery/jquery-storageapi'
], function ($, Component, customerData, _) {
'use strict';
return Component.extend({
defaults: {
cookieMessages: [],
messages: []
},
/**
* Extends Component object by storage observable messages.
*/
initialize: function () {
this._super();
this.cookieMessages = _.unique($.cookieStorage.get('mage-messages'), 'text');
this.messages = customerData.get('messages').extend({
disposableCustomerData: 'messages'
});
// Force to clean obsolete messages
if (!_.isEmpty(this.messages().messages)) {
customerData.set('messages', {});
}
// this line clear the current messages.
$.cookieStorage.set('mage-messages', '');
}
});
});
Issue is messages.js must be called multiple time in homepage. thus the last line of the above code clear out the messages every time. you have to find out why this file calling multiple times in homepage.