I think I have managed to get it working with a custom UrlProvider and ContentFinder:

public class SiteUrlProvider : DefaultUrlProvider
{
    public override IEnumerable<string> GetOtherUrls(UmbracoContext umbracoContext, int id, Uri current)
    {
        var urls = base.GetOtherUrls(umbracoContext, id, current);

        var content = umbracoContext.ContentCache.GetById(id);

        if (content?.DocumentTypeAlias == MySite.ModelTypeAlias)
        {
            // add the node ID to the URL, since this shouldn't be the home page URL
            urls = urls.Select(u => $"{u}{id}/");
        }

        var homeNode = content?.AncestorOrSelf<MyHomePage>();

        if (homeNode != null)
        {
            // remove the home node url name, since this should be the home page URL
            urls = urls.Select(url =>
            {
                string path;

                if (Uri.IsWellFormedUriString(url, UriKind.Absolute))
                {
                    var uri = new Uri(url);
                    path = uri.GetAbsolutePathDecoded();
                }
                else
                {
                    path = url;
                }

                var newPath = path.TrimStart($"/{homeNode.UrlName}/").EnsureStartsWith("/");

                url = url.Replace(path, newPath);

                return url;
            });
        }

        return urls;
    }

    public override string GetUrl(UmbracoContext umbracoContext, int id, Uri current, UrlProviderMode mode)
    {
        var url = base.GetUrl(umbracoContext, id, current, mode);

        var content = umbracoContext.ContentCache.GetById(id);

        if (content?.DocumentTypeAlias == MySite.ModelTypeAlias)
        {
            // add the node ID to the URL, since this shouldn't be the home page URL
            url += $"{id}/";
        }

        var homeNode = content?.AncestorOrSelf<MyHomePage>();

        if (homeNode != null)
        {
            // remove the home node url name, since this should be the home page URL
            string path;

            if (Uri.IsWellFormedUriString(url, UriKind.Absolute))
            {
                var uri = new Uri(url);
                path = uri.GetAbsolutePathDecoded();
            }
            else
            {
                path = url;
            }

            var newPath = path.TrimStart($"/{homeNode.UrlName}/").EnsureStartsWith("/");

            url = url.Replace(path, newPath);
        }

        return url;
    }
}

And the ContentFinder class:

public class SiteContentFinder : IContentFinder
{
    public bool TryFindContent(PublishedContentRequest docRequest)
    {
        var contentCache = docRequest.RoutingContext.UmbracoContext.ContentCache;

        string route;

        if (docRequest.HasDomain)
        {
            var domainNode = contentCache.GetById(docRequest.UmbracoDomain.RootContentId.Value);

            // get first child node of domain node, as that will be the home node
            var homeNode = domainNode?.Children.FirstOrDefault();

            // include url name of the home node in the route path
            route = docRequest.UmbracoDomain.RootContentId.ToString() + (homeNode != null ? $"/{homeNode.UrlName}" : null) + 
                        DomainHelper.PathRelativeToDomain(docRequest.DomainUri, docRequest.Uri.GetAbsolutePathDecoded());
        }
        else
        {
            route = docRequest.Uri.GetAbsolutePathDecoded();
        }

        docRequest.PublishedContent = contentCache.GetByRoute(route);

        return docRequest.PublishedContent != null;
    }
}

Register them in the ApplicationStarting event before the defaults:

protected override void ApplicationStarting(UmbracoApplicationBase umbracoApplication, ApplicationContext applicationContext)
{
    // register custom URL provider
    UrlProviderResolver.Current.InsertTypeBefore<DefaultUrlProvider, SiteUrlProvider>();
    ContentFinderResolver.Current.InsertTypeBefore<ContentFinderByNiceUrl, SiteContentFinder>();
}