But she is far more comfortable looking away while in conversation. The theory that has resonated most with me is that both facial interpretations and language processing are very cerebral processor intensive. Turning off your facial tracking algorithm by looking away frees more brain nodes to help with processing and thinking about what you are hearing and saying.
So I’m torn. Do I continually push her to look at faces during conversations, because over time it will build up more efficient neural pathways for that kind of processing? Or is it just something unique about her I can’t change and I am just causing her unresolvable stress?
Why don't you let her decide?
I'm sure you asked this mostly rhetorically, but I thought I'd chime in with my two cents.
If your daughter is on the autism spectrum (guessing from the context) and is 13, she probably already finds socializing somewhat aversive and exhausting. Pushing her to make the experience even more aversive by insisting she look people in the eye is probably going to make things worse.
But I don't think that means that you can't change it. I would just suggest practice eye contact or looking at faces in small intervals only during some conversations. E.g. have a schedule and do a short conversation once a week where she deliberately practices it.
More generally, maybe try to think of the goal as moving the needle from where she is, not to achieve neurotypical levels of eye contact.
I've also had the eye looking problem, and yes, it was uncomfortable and harder to concentrate. But now it I'm around 30 it has become quite natural. Not a parent, but what I imagine might help besides 'just trying to do it' is seeing the use and fun of eye contact:
* Explain eye contact helps people understand you are interested in them. Of course hopefully you have some real, friendly interest in someone ;-). Friendly listening to someone and looking at them is a good step to getting more familiar/friends.
* Maybe help her recognize more of the expressions in and around the eyes. It is a very interesting part of the face. And you don't have to look someone exactly in the eye to see most of it... Besides angry/sad/happy, what are the little signs that show someone is becoming irritated, lost interest or getting bored, is thinking, is messing with you, disagrees (but maybe too polite to tell), etc. Recognizing it in movies, and possibly imitating the expressions can also be a 'safer' step.
* It works two ways. You let others see more of you by looking them in the eyes. You have to open up in a way, can get more self-conscious, and thus uncomfortable. Looking someone in the eyes can make them trust you more, helps them see your intentions, if you really mean what you say, that you know what you are saying... But you may just want to start opening up some more with people you already feel comfortable with.
Still, eye contact is culture dependent. For me, what helped me were these motivations, and that I forced myself to have on estimate 100+ little wel-intentioned conversations with strangers per week (Jehovah's Witness).
What you describe happens to me especially in groups. My trick is to pick a point slightly off - their forehead, a little to the side, a random visible point just past them - and focus there. It's almost imperceptible to most especially if they're caught up in the convo, and allows me to process. Sometimes in a long convo, I may have to change my focus point multiple times. It takes practice, and you only get that actually talking to people.
Another thing to keep in mind depending on severity. Depending on her niche (the 1 or 2 things her brain specializes in) social interaction over an extended period will make her mentally tired. In a large group, this can happen exponentially.
I do okay one on one. Or in groups of up to half a dozen or so. However, I built my coping mechanisms over 30+ years. Even then, I have to pace myself, for example if I am at a party. I've spoken to large groups before as part of my day work and volunteer work. 2-4 hours at a party or 20-60 minutes in front of 100+ people, and I am exhausted. This is because the part of the brain I'm using for that task in me wasn't built for that.
If you want an analogy, imagine the hardware bits of a computer for an analogy of my brain. Now imagine all the parts are wired together differently from the standard architecture of such a system. The GPU is being used for CPU tasks, so other part is being used for a purpose not originally designed, and so on. The OS has been written to compensate for the architecture change but there's no documentation and the person who wrote it has vanished. Topper, the video and audio outputs instead of going to a monitor and speaker are hooked instead to a blender full of margarita mix. Basically, that's my life.
That doesn't mean she can't make friends and have a fulfilling life. It doesn't mean she doesn't feel as much as others. If anything, my emotions are probably more intense than a lot of folks. Aggravated by not being able to express them effectively. The friends you do make and keep mean that much more, especially if they know what you are. If also means she may feel her failure to keep some friends more deeply. I have kept friends through adulthood and a war or two. I've also lost some and I will never stop missing that particular part of my soul.
Try the slightly off focus point trick, give it time and practice.
(FWIW, the lock down has limited my social activity to circles that tolerate my eye contact aversion, and I am taking advantage of it to get what slack I can)