I used to rank on the first page for hundreds of high quality single and two word phrases related to my industry. Since May, the only way I am able to come up for these phrases is if I include "keyword" in the phrase. So if my domain is keyword.com and the phrase I used to rank on is "widgets", the only way to see my site is to search for "widgets keyword".
It could be a lot of different things of course, but I'll venture a guess and say it's an inbound linking problem.
An additional word will of course make the search much less competitive. In this case, since it's your domain name that's providing the boost for "keyword", it's likely that you've maintained good inbound linking for the domain name but have lost some inbound linking reputation on the other competitive words on which you previously ranked. It's likely that the addition of your keyword to the search puts you territory where you still have an advantage.
Keep in mind that my domain is a common term.
As you describe the situation, it sounds like it's much less common or competitive than you suppose. I'd trust the numbers of results returned on the Google serps page to provide at least a very rough indication of how competitive these searches are. Run the numbers and see. I'd run lots of different combinations... but basically I'd compare...
[widgets keyword] with [widgets]
allintitle: widgets keyword
I've also seen pages that have a word or two from the domain or company name in the query sink like rocks when a very competitive modifier was added to the query. They simply didn't have enough inbound authority for the additional term to rank for it... analogous to a word in your "widgets" phrase.
So the question may be, what did Google change to change the effect of your backlinks? What was the quality of those backlinks, and are they counting now?
And/or, it may be, how are the backlinks now meshing with your pages?