How does one Render full html field from a field_target_entity ?

Thanks @sutharsan.

Your advice #2 worked for me when everything else failed.

You made my day. :)

I struggled with this thing for a while, and while this helped, I realized that I can do something similar with "view" filter from twig_tweak module.
That filter will extract a render array from provided field item that will look similar to what @sutharsan provided.
E.g. in a twig for a field that is a reference you can display a formatted long text like this:

{% for item in items %}
  <div>{{ item.content['#entity'].field_name.0|view }}</div>
{% endfor %}