The Emoji you are referring to (https://emojipedia.org/pleading-face/) is unfortunately not yet supported in the Unicode version the add-on uses.
However, support for more new Emojis is planned and tracked in https://github.com/rugk/awesome-emoji-picker/issues/34. It will be released in one of the future updates, but we depend on other developers (upstream libraries) to make this happen. See the GitHub issue for more details.
Thus it may unfortunately take some time.

If you do only see two-letter codes for the country flags, this may be a limitation of your operating system.
If you use Windows, Windows does not have flag icons for all these countries currently.
However, you can solve that by choosing another emoji style. To do that, right-click on the icon of the extension, choose “Manage extension”, “Settings” and choose a different icon set next to “emoji set”.
By default, this extension uses your system emojis, but there are different styles available and they all have proper emoji flags.

If you have further issues or questions, feel free to ask or suggest a feature by opening an issue on https://github.com/rugk/awesome-emoji-picker/issues/. Thanks.

If this helped, it would be glad if you could adjust your voting.