They've decided together that he's not going to step in for sick days if it doesn't suit him because they're both choosing to prioritise his job.
OP is exhausted, catching every big going, probably about to be fired and has a poor sickness record. Jobs are thin on the ground everywhere. I'd say her chances of getting employment elsewhere, quickly enough to not be fired, are pretty much zero. She's also going to be in the same situation pretty much wherever she works. She's exhausted so will continue catching things for a while until her health picks up. She's got a little bit of the mentality of not going to work with a sniffle too (it's obvious from her posts). So she's not realistically going to be wanting to send sick kids to school or go into work herself with a cold, because she doesn't believe in it. It's a mentality that leads to increased sickness absences than the average person.
Yeh being a SAHM is a risk, but you can't avoid all risks in life. They're married. That gives some legal protection and a backup plan of a divorce settlement. It also means they're supposed to be a team, supposed to trust each other and put their faith in each other. If they're not going to do that then what was the point of getting married in the first place?
Why have kids with a husband you can't trust? May as well just get pregnant from a ONS and not tell him, less hassle. The whole point of using marriage as the basis for having kids is that you can supposedly trust each other and lean on each other in times of need. Like when you're raising little ones. If you can't or won't do this, there's really very little point in it at all and it becomes nothing more than a chance to dress up and play at being "princess" for a day while being fussed over by everyone.
This "let's be hyper-independant just in case" mentality is destroying women's lives. It's created no less of a precedent than the one you mentioned of becoming the default parent. It has created situations like the OPs, where she's run ragged and whatever she does is wrong in someone's eyes so she feels guilty constantly, contributing to her stress levels and weakened immune system that's left her floored.
Who cares about becoming default parent? That's exactly what SAHP is. And when they're no longer the SAHP a conversation can be had about the default parent status needing to change, which also means the person who was default parent taking responsibility for their own actions and refusing that role when others try to place it back upon them. It really shouldn't be a huge thing. Life situations change and so does the role various family members play in the family dynamics.
Everyone goes about these days basically acting single and child-free, except they're married with kids! Then wonders why life isn't working out so well and they're dissatisfied.
There's always a risk the marriage will end. That's life. It doesn't mean you go through life acting like it's definitely going to happen. If he leaves her she can worry about that then.