Hi Robert, sorry for my confusing post. I'll try to clarify.
I did a plain vanilla search of our domain name with nothing else.
So in this case, I just typed:
We do show up 1st in the results, and then the next 20 results shows results from the big brand site:
There are no hyphens in the domain name. After page 2, on page 3 and deeper, we then see a few more results from our domain.
The thing is, you mentioned that the other results (not from my domain) will show because they might mention our domain, but in the results (all showing from the brand site) non of the results mention our domain at all. instead, you know how google bolds the searched keywords? In this case, they bold all the text showing the big brand site's name.
So all the text of widgets.tld are bolded, almost as if google is assuming that's what we searched for, instead of widgetsarefun.tld. Then on deeper pages (3+) it shows results that contain our domain name mixed with other results showing a bit more variety.