If you want to play their games, you need to have their credentials. It is extraordinarily difficult to try to maneuver around that. Professional jobs are 100% "their games". Illegals are not competing in the game, they are essentially NPCs (purely in the video game sense) and cannot have upward mobility, if that even remains a thing.

If you want to live on the periphery, and are willing do the hard work of figuring out how that life could work and raise your kids in that life, then it is more feasible. You mentioned Amish elsewhere, and that is a decent model. Plenty of them have the documents and play by the rules where they need to interact with government, banks, etc., but there is also room for plenty of their population to not need that stuff.

I thought deeply about this when we had our kids, but we're older and I couldn't convince my wife to do home births (we didn't vaccinate and are homeschooling though).

I think a better thing to research is how long can you delay it and not have problems (if you are going to do a homebirth). Try to find examples of kids getting birth certificates later on (1 year, 5 years, maybe 15 years). Is there some way you can claim "I meant to do it and then got distracted with life and forgot about it for 15 years"? That would buy you more time and put off the decision until the last possible moment (the point at which they need it if they are going to enter society).

If you do that, maybe document the birth event somehow, with some makeshift certificate or something that indicates "I documented the birth happened at this time" so you've got something to show if they question things. Then claim ignorance and forgetfulness. Again, though, before you go down that path find some clear legal foundation that shows it is possible to get one at some later time.