I have found a fix after looking at ContentTreeController class and comparing the difference between my custom implementation. I found that ContentTreeController had this attribute applied to the class [SearchableTree("searchResultFormatter", "configureContentResult")] and after applying this to my implementation, it sttarted showing the URL in search results.

Full implementation is below for anyone wanting to replace umbraco backoffice content search. It will search content by node name and url of content.

https://our.umbraco.com/Documentation/Extending/Section-Trees/Searchable-Trees/index-v7/

    [SearchableTree("searchResultFormatter", "configureContentResult")]
[Tree("content", "content")]
[UmbracoApplicationAuthorize(new[] { "content", "media", "users", "settings", "developer", "member" })]
public class ContentSearchController : ISearchableTree
{
    public String TreeAlias => "content";
    public IEnumerable<SearchResultItem> Search(string query, int pageSize, long pageIndex, out long totalFound, string searchFrom = null)
    {
        var UmbracoHelper = new UmbracoHelper(UmbracoContext.Current);
        var RootNodes = UmbracoHelper.TypedContentAtRoot();

        var SearchResults = new List<SearchResultItem>();
        var MatchingItems = new List<IPublishedContent>();

        foreach (var RootNode in RootNodes)
        {
            var Results = RootNode.DescendantsOrSelf().Where(x => x.Name.ToLower().Contains(query.ToLower()));
            Results = Results != null && Results.Any() ? Results : RootNode.DescendantsOrSelf().Where(x => x.UrlAbsolute().ToLower().Contains(query.ToLower()));
            if (Results!= null && Results.Any())
            {
                MatchingItems.AddRange(Results);
            }
        }

        if (MatchingItems != null && MatchingItems.Any()) {
            SearchResults = MapSearchResults(MatchingItems);
        }

        totalFound = MatchingItems != null ? MatchingItems.Count() : 0;

        return SearchResults;
    }

    private List<SearchResultItem> MapSearchResults(IEnumerable<IPublishedContent> MatchingItems)
    {
        var SearchResults = new List<SearchResultItem>();

        foreach (var MatchingItem in MatchingItems)
        {
            int ParentID = MatchingItem.Parent != null ? MatchingItem.Parent.Id : -1;
            string Icon = ApplicationContext.Current.Services.ContentTypeService.GetContentType(MatchingItem.ContentType.Id).Icon;
            var SearchResultItem = new SearchResultItem();
            SearchResultItem.Id = MatchingItem.Id;
            SearchResultItem.Alias = MatchingItem.DocumentTypeAlias;
            SearchResultItem.Key = MatchingItem.GetKey();
            SearchResultItem.Name = MatchingItem.Name;
            SearchResultItem.ParentId = ParentID;
            SearchResultItem.Path = MatchingItem.Path;
            SearchResultItem.Score = 1.0F;
            SearchResultItem.Trashed = false;
            SearchResultItem.Icon = Icon;
            SearchResultItem.Udi = new GuidUdi("document", MatchingItem.GetKey());
            SearchResultItem.AdditionalData["Url"] = MatchingItem.UrlAbsolute();
            SearchResults.Add(SearchResultItem);
        }

        return SearchResults;
    }
}