How to create app-like Chrome shortcuts?
You wait for the web app developer to implement Chrome's WebAPK on Android feature/requirements.

From the following post: How to Remove Chrome Logo from PWA App Home Screen Link (Android O Preview)

Digging through the comments on the accepted answer from Anand dated Dec 17, 2018:


  If you are asking how to avoid chrome icon, as long as your went [sic] app is fully qualifies PWA to be installed as APK, new versions of chrome won't add that icon. I've tested with chrome version 70 in Android pie . If it's a mere web page, failing to meet key PWA criteria or non PWA, chrome might add chrome icon to indicate it's a web link(depending on chrome version again)


Further research on details led me down a rabbit hole of what features exist on Chrome vs Android and the meaning of 'install' to home screen, which while it does somewhat affect end-user behavioral expectations, the implementation details are off-topic for this site.

If you can live with icon badging, you may want to try your site/webapp with Firefox for Android and see if the app UI still retains the undesirable "header".