Op, you didn't do yourself any favours with the wording of your original post, I think that is the main issue here.
"At the very minimum, everyone should have an emergency fund - yet plenty of people live pay check to paycheck with no backup.
AIBU to think that saving should be non-negotiable? Or do some people genuinely have valid reasons for not doing it?"
Those two lines are the ones people are taking offence to. You say in a future post that saving may not be realistic in that moment but in your first post say EVERYONE should have an emergency fund and "people live pay check to paycheck" well that's sort of the point the ones living like that are the ones who have zero chance of saving.
You then state "do some people genuinely have valid reasons for not doing it"
This comes across as very out of touch. Maybe if you had phrased it differently and said something like "While a lot of people are in a position that they financial cant, why do you think those who could choose not to". While not perfect that at least reads like you have some understanding.
But to answer what I think you are trying to ask, a very large group of people in this country do not have a choice to save, they either eat or save, what's the point in saving if you cant eat? Another group of people could save but it would cause a drop in their living standards, they would need to cancel netflix or not go out for dinner that month, they decide that happiness now is worth more than security in the long run (I fall into this group). I know the risks I am taking but its a trade off, I either live a bleak life now without much in the way of "fun spending" or I save for a potential future need. My choice is who knows what the future holds, I would rather enjoy life in the moment and figure out the future, you never know if you're going to make it.
Hopefully one day I will be back in a position to do both, but while I cant the present takes priority.