I noticed the other Google App Script Web App that was working had a url structure different than this one with issues...

Problematic Web App URL: https://scripts.google.com/macros/s/{script_id}/exec

I tried to publish another app, initially with "Google Account logged in" can access the web app (not Anyone can access). When I published with this permission, the generated deployment URL was in different structure, like this (https://scripts.google.com/a/macros/{script_id}/exec), then I changed the permission to "Anyone" can access... and I tried to execute embed the web app with this new URL structure and it worked.

So What I understood, if anyone face this issue, must use different url structure to get the url with /a/macros/... instead of /macros/s/

So this URL: https://scripts.google.com/macros/s/{script_id}/exec (will cause issues if multiple goolge accounts logged in on mobile chrome browser...

and this URL: https://scripts.google.com/a/macros/{script_id}/exec should be working without URL changes.

For me it worked and saved my many hours of work on migrating web app to alternative platforms.