I used to think this way, and this line of reasoning misses the point.
You don't ask someone what they had for lunch to find out what they had for lunch. As you said, that piece of information has no bearing whatsoever for you. I may even ask someone what they had for lunch already knowing what they had for lunch.
The reason you ask what they had for lunch is that doing so is an explicitly social behavior, or as you call it, "keeping up appearances". The fact that you're willing to do the song and dance just like everyone else means you are sincere about keeping peace and order within the social group you're interacting with. You understand there are certain expectations, and you fulfill them. That's the behavior of someone that cares.
If you were truly insincere, you wouldn't bother keeping up appearances. You would tell that person "Oh fuck off with your lunch, no one gives a shit".
I've come to the same conclusion. I've developed a mode that I switch on when needed. It is typically engaged when charming/schmoozing/selling/persuading is involved.
> But I feel somewhat insincere doing
I don't think you should. You work with humans as well as machines (I'm guessing). Some of these humans crave social activity and respond positively when they get it. That could mean telling someone about your day. It could also mean listening about their day.
> that and it takes a lot of effort emotionally.
I hear you. I burn energy very quickly when my extrovert mode is switched on. What's even harder, for me, is switching back to introvert related work after being in extrovert mode.