The study found that people are less successful at spotting logical fallacies when the conclusions are supported by their politics.
This is itself an interesting finding, but (1) I don't think the "everyone has biases" argument is justified by this article and (2) that precise argument is often used by people to apologize for allying themselves with others who have abhorrent views but are otherwise inline with their political interests (e.g. White Evangelicals in Iowa who vote for Steve King). I'm 1000% not suggesting that you're doing this but I think it's an unfortunate conclusion many draw from your argument.
Yes everyone has some level of irrationality but not everyone has a sound ideology, many people often waiver between conclusions and human judgement is real, important, and helpful.