Hi,
these are two different topics, let me address them separately:
Icon with white background:
Yes, I know that quite a few launchers display icons this way. These launchers expect apps to produce shortcut icons following the "Adaptive Icon" pattern. In general this is a good idea - however, Native Alpha cannot use this approach without editing the PNG icon the website is providing and that's not something I looked into yet (and won't in the near future). Currently there is no established standard for website to provide icons in a way it would work as adaptive icon.
TL;DR: Use another launcher, I can recommend open-source Lawnchair but there are many others.
App badge:
This is another topic which the app has no control over, it is decided by the launcher if shortcuts show their parent app's icon in the right lower corner. A Native Alpha user on Reddit suggested "Smart Launcher" as alternative which does not. I don't know any other one. Afaik Google recommends this badge as security feature, so apps cannot easily produce shortcuts in order to pretend to be other well-known apps, e.g., pretending to be your banking app.
Thank you for all your efforts!
The home screen icon created is a small square set against a white background.
Additionally there is always a native alpha icon badge to the bottom right of the icon.
Is there a way to have a full sized icon images and no badge displayed?