Counts give you a good hint about what you'll see when you start building formal statistical models. But counts are NOT those formal statistical models, so we don't use the results use the results of their stat tests to make big decisions. I would include that attribute in your HB analysis and use those models to make decisions about the final structure of your model.
It sometimes happens that an attribute doesn't look like a significant predictor because of respondent heterogeneity. For example, say an attribute is color, orange or blue. If half your respondents love orange and hate blue while the other half hate orange and love blue, then in the aggregate analyses (counts, logit) it will look like color doesn't matter. When you run HB you might see that orange and blue still have (average) utilities close to zero but that color had a significant level of importance. This is because you have a lot of blue-lovers and a lot of orange-lovers and all that love cancels out in the aggregate. If you don't include color in your HB model you'll never catch this.
Of course it could also happen that color isn't important in the aggregate and then it isn't important at the respondent level either. Then I suppose you could drop it from the HB analysis, but most folks would probably still keep it in because if it's that unimportant it is unlikely to change things by being left in, while it might raise questions if you leave it out.