Lazy loading is not loading due to the JavaScript errors caused by sumo app and one more mini cart app, resolving this will more likely fix the lazy loading.
To improve the code further, we will have to minify the files, strip away the unwanted excess codes like CSS and js, combine the different code files together and optimize the codebase. To optimize the app we can code the custom function right in the theme template or choose alternative apps that don't affect the speed. As we have major issue with above mentioned JavaScript error we can’t exactly figure out where to improve, as of now I can get the lazy loading fixed and the other first tier tasks to improve the page load speed.
I have good knowledge in the LIQUID code that Shopify stores are built with and HTML, CSS, javascript, and API, hence I can increase the page speed without affecting the existing design and the functions. And I won’t use any app for this as you requested, everything will be 100% native Liquid code.