Birth control is available over the counter in the US. If you have specific need, it’s very cheap and your doctor will almost always just call in refills without charging you. The meds themselves are not expensive.
Annual visits are also not actually that important. They’re perfunctory. They can certainly be put off for a few months in a ok except a few one in a million cases.
And at any rate, as I said, losing your job in the US means your insurance is disrupted, not that you are now uninsured. Pregnancy even in states without Medicaid expansion will get the mother and child on Medicaid.
Of course, at the risk of being silly, it’s also true that missing a day of birth control is not what got your wife pregnant. ;) it’s pretty surprising to me how many people (now with children) thought birth control meant they wouldn’t get pregnant. There definitely needs to be better education on this. Taking birth control, even regularly, even with an IUD, is more like a backstop and should not be relied on for your primary protection. The odds are low but when you play them a few times a week for ten years…