Yes, the Community user definitely does bump these questions. I've found one on Meta Stack Overflow that has a zero-score answer and a status-completed tag but was bumped by Community after the tag was added.
I haven't looked through them all but using the following search, you should be able to find more of them on any meta site you like:
[status-completed] isanswered:0 answers:1
This returns questions with the status-completed tag, questions that, by the system's definitions are not answered (have no answers with a score greater than 0) and do have at least one answer.
An added caveat is that if an answer is accepted it isn't bumped by community. As an example, this question. So, to filter these out of your search results, you can add "hasaccepted:no" to your search terms. This leaves only 35 questions on MSE, actually (down from 200). A short enough list to look through.
You can further drop this number to 10 by adding "closed:no"... making me wonder if this is really a big enough concern to be worried about. Of these ten questions, none of them qualify to be bumped by Community as all of the answers have negative scores.
Now, some sites may use the status tags more than MSE... but I'm guessing not. Even MSO only has 9 questions that meet the final search:
[status-completed] isanswered:0 answers:1 hasaccepted:no closed:no
I suppose it's possible that I've erred in my search terms, so I'm happy to be corrected if I've done so.