This could also be a methodology of solidifying an uncompromising bias.

The "given" was a pretty near unchangeable opinion. So, the chance for further solidification isn't something to be concerned about. My only point here was that even in such cases, there's still some utility to debating with people on the other side just to hone your arguments (not to further determine the objective truth). But if we do want to still assume a chance for a change of opinion, debating will increase the likelihood compared to just not discussing the issue at all, especially if you try to find the best presenters of the opposing side's ideas.

Unless you've hopped down the rabbit hole of solipsism

I rejected that rabbit hole very early in life :-)

"does it really matter if my belief is biased? How does it affect the trajectory of my life and the life of others?"

If I understand you correctly, you're suggesting that bias that leads to wrong conclusions may not be something to worry much about if it doesn't have a negative impact on our life. If so, I guess that could be considered something of a truism on a utilitarian basis, but if our passion is just for truth independent of utility, it obviously still matters. And I'm not sure if we can always correctly measure the impact of mistaken beliefs based on biases either.

I also like to draw upon the game of poker where my constant mantra is, "does the story make sense?" Mix that with a dash of Occam's razor and two dashes of Hanlon's razor, boom, you have my personal bias cocktail.

I think that's not a bad formula for evaluating human behavior and it share some similarity to my own. I wasn't familiar with the term Hanlon's razor, but as a concept I'm a strong believer in it and frequently use it when trying to understand human behavior.

show that you have given consideration to their perspective without being blatantly disrespectful.

Yes, completely agree with this. It can be hard at times, but if you really want to change someone's mind, I think its critical.