Do children need visas to enter Thailand if they hold US passports and are under 14 years old?

Or, another possible option: talk with your child's hospital and doctor about whether your child can apply for a medical extension. Medical extensions allow caretakers (not for only one if I remember right, but not sure how many max). you can talk with them (and the local immigration office) if this extension can be applied to your whole family.

Doesn't matter where the kids were born, matters what passports they have and you say they're all holding american passports and your wife holds a canadian one

If you live in Mexico and you're going to get visas you get them at the Thai consulate in Mexico City. They are not in the eVisa system so you apply at the consulate for the visas

You can all show up here without visas and get stamped in free for 30 days on whats called a 30 day visa exempt entry. That can be extended one time for 30 more days at the immigration office for 1900baht..

The rule is children under 15 cannot be fined for overstay and children under 18 cannot be banned. There is no negative repercussions to the child or the parent and it will not cause any issue for them or you comin' back in to thailand

I know foreigners here with foreign children on multiple years overstay who exited/re-entered without any issue.

Just bring 'em in and let 'em overstay would be my advice especially seeing as they're really young children.