Workplace Strife Advice

This post is a second chance.  A respected coworker asked me for advice on a heated personal office conflict, and while I think I gave a good answer, he was unconvinced.  In hindsight, I did not do a good job explaining some good reasons for the answer I gave, which summarized, would be:

Take the easy out and move on as quickly as possible.

While I suspect there is professional pride involved in the disagreement, it is in the interest of everyone in the office to get over it and put it all in the rear view mirror so that work can progress.

My second chance advice on the nature of the conflict derives from what I know about my coworkers:

• They are both experienced team leads
• Both teams work on the same software project; developers and quality control
• They work in geographically separated offices 
• Neither has met the other in person
• They do work together on a daily basis via team chat, emails and morning scrum

I have an long drive home from work, so I naturally started to reflect on that conversation and the difficult situation that my friends were in.  I came up with the following points that may help someone else who is experiencing a difficult personal interaction that is seemingly at an impasse.

The very best solution, of course, would be for both people to meet (or call because of being on a distributed team) and work through the differences so they can continue together as a high functioning team.

But short of that, I believe all of the points below are helpful.  They are not meant to be used all at once as a big fire-hose solution to extinguish a workplace fire.  Rather, I'm hoping one or more of these register as a well made point that makes sense and helps someone through a similar conflict.

I also hope these points are not taken as being callous, as any one of them would resonate with me if I was in the same situation (seemingly stuck without chance of agreement nor common ground reached), and someone offered me an outside view of my situation.

1. Your future self does not care about this argument as much as you do. Look back on this situation in 10 years and realize that it simply was not that important.
2. As a team lead, you need to consider that time spent extending a conflict is time taken away from your leadership responsibilities owed to your junior team members
3. Being wrong can be a drastic side effect of choosing your Perfection over their Very Good.
4. The thing you are arguing over may be of-value, but the argument itself is valueless. You were hired for your life experiences and professional skill set, neither of which are being put to use to solve the problems you should be working on. You expect to be paid for excessive time spent arguing?
5. So, you've been challenged to a High Noon showdown and you're asking for advice on which ammunition to use instead of how to avoid a public gunfight?
6. If you're mutually blocking each others progress, does it matter who gets credit for standing their ground the longest? There is no absolute right or wrong in matters of opinion.
7. Don't get blocked! It IS within your power to move forward. 
8. Is your professional ego more important than your human interactions? What is the point of being "right" if you are causing, or prolonging, a painful situation when both of you are literally losing sleep over it? Conflict is natural, but suffering is not.

What advice would you give a coworker in a heated, ongoing situation?