People are better at identifying biases in other people than themselves, because when we look at others we see their behavior, but when we look at ourselves we have what seems to be a better tool - introspection. We can explore our own minds, examine our own motivations, and thus determine if our motivations are appropriate. However, in reality introspection is actually a very poor method for highlighting bias, our own biases are a major blind spot for introspection.
The best way to identify your own biases is to step outside of your own head, and to try and imagine someone else doing exactly what you are doing. Would they appear biased to you? The answer may be more accurate than the normal approach.
Introspection does have the weaknesses you describe, but I think it is possible to develop more skill at honest introspection with practice. And introspection has one big advantage, in that its readily available at all times.
But if a person relies solely on introspection, I have to agree that they will inevitably go down some completely wrong lines of reasoning along the way, no matter how brilliant they are. During my college days, I remember reading an argument by Aristotle in which he made a completely wrong leap of reasoning that was only obvious to someone with modern scientific knowledge. And I remember some early "guesses" about science that I had that were completely wrong for similar reasons. Having external critics brings extreme benefits to logical thinking.
This is actually very similar to a technique I like to use when trying to persuade someone. People are often biased in favor of the ideas or actions of politicians they support, so when I want them to critically examine the behavior or ideas of that politician, I suggest they try to imagine how they would view the action if it was committed/suggested by a politician they dislike.