This common use of "I know I'm biased, but..." and its equivalent phrases is definitely a good thing to point out and work to avoid.

The proposed catch-and-analyze method for when you say such things yourself would also be useful from the other side of the conversation, as a more explicit exercise: Your conversational companion says 'I know I'm biased...' and that's a signal right there for you to ask 'how/why?' and get them thinking and talking about it. I actually think that done right, it could be turned from an unproductive 'please ignore my bad argument' signal into a pretty good jumping-off point for a double-teamed analysis of the issue.

In this vein, I actually find it useful to state my biases sometimes in conversation, as a sort of assisted sanity check - my friend might be able to catch connections from some of those biases to my arguments better than I, and in stating them, I explicitly remind myself what they are and that I should be dealing with them. If for instance I'm biased in favor of Idea X by my Inherent Trait Y (e.g. something like being a student, white, female, etc) Trait Y (hence the potential for bias) isn't going to go away; therefore the most productive path is to a) acknowledge it and b) apply a correction factor to the weighting of arguments that link to that bias.