You need to scan and replace the HtmlFragment. Checking if it's an image and then loading the alt text and replacing it. I've knocked up a demo no alloy that works, just use this extension instead of ToHtmlString when to get the string of the XHtmlStirng for your ViewModel / In View. Replace the namespace in this extension class with whatever your project is.

using System.Text;
using System.Web.Mvc;
using EPiServer;
using EPiServer.Core;
using EPiServer.Core.Html.StringParsing;
using EPiServer.Web.Routing;
using FrontEnd.Models.Media;

namespace FrontEnd.Business
{
    public static class HtmlStringExtensions
    {
        /// <summary>
        /// Parses the XhtmlString for Image tags and sets them to lazy load
        /// </summary>
        public static string FormatHtmlString(this XhtmlString html)
        {
            var stringBuilder = new StringBuilder();

            var altText = string.Empty;

            foreach (var htmlFragment in html.Fragments)
            {
                var value = htmlFragment.GetViewFormat();

                if (htmlFragment is UrlFragment urlFragment)
                {
                    if (!string.IsNullOrWhiteSpace(urlFragment.InternalFormat))
                    {
                        var content = UrlResolver.Current.Route(new UrlBuilder(htmlFragment.InternalFormat));

                        if (content is ImageFile imageData)
                        {
                            altText = imageData.Description;
                        }
                    }

                }
                else
                {
                    if (value.Contains("alt=") && !string.IsNullOrWhiteSpace(altText))
                    {
                        var indexOfAlt = value.IndexOf("alt=\"") + 5;
                        var startString = value.Substring(0, indexOfAlt);

                        var restOfString = value.Substring(indexOfAlt);
                        var closingIndex = restOfString.IndexOf("\"");

                        var endString = restOfString.Substring(closingIndex);

                        value = startString + altText + endString;
                        altText = string.Empty;
                    }
                }

                stringBuilder.Append(value);
            }

            return stringBuilder.ToString();
        }
    }
}