How do I become a primary caregiver?

Ok SPC, the unfortunate nature of this entire situation is the fact that your spouse is currently in NY and you are in NC. As an active duty service member, your spouse is assigned a PCM at your local MTF and would require an ETP to deliver outside the network. You are authorized 21 days of Secondary care giver leave regardless if she has a C-Section or not. (My wife delivered via C-Section) The kicker to this leave is that it is Local leave and you cant just decide to take it elsewhere. Although its considered a "Major Surgery" due to the nature of pregnancy, her abdominal muscles have already separated leading which is actually somewhat beneficial. My wife was able to walk around, lift our son and do most things 3 days post op. 21 days is more than enough recovery time hence why MEDCOM upped the leave from 14 to 21 days to factor this in.

If you don't receive clearance to deliver outside your assigned MTF, you will foot the bill for the procedure.

I was a full-time caretaker for about a year as my wife's health cratered. Whenever she was hospitalized, there were 4-5 folks tending to her needs. At home, it was me. A traveling RN visited for about an hour weekly, eventually, the OT & PT folks would come in for an hour twice a week.

I can offer my support - from a distance - during your adventure. My son was a C-section, born while we were in Wiesbaden Germany.
How the Army presently deals with family crisis situations of this nature is completely different than it was in my day, so I'll not venture into that topic.

This is actually a great question. No where in the regulation or MILPERs have I seen guidance for a family when the father is the only military member. It does not specify what the requirements for being a primary caregiver are.

I have seen a father be primary in a dual military couple though, but never when only 1 is military.