Lots to unpack there, but I'll give it a shot. One conservative Christian man to another...in front of everyone.
A: The covid shot requirement does not take military aviation off the table for you. It might right now, but not forever. Eventually either the military will withdraw the requirement, or there will be another option of non-stem cell based vaccination (like an actually researched and studied non-emergency-use vaccine should be). It all depends on your basis for rejection. I, for one, had no defensible basis, as I happily and ignorantly received 8 rounds of the anthrax vaccine without asking twice, as I was young and stupid. That proved to be 100% risk/0% reward. Long story short, make sure you sincerely understand your basis for refusal, and that it's not just under the "I will not comply" category just because you don't like it. That's a childish defense based on ignorant emotion. If you have a sincere objection based on logic, stand firm on it, and find another way to your dream.
If you say "it was my dream..." and hold that with the same hand as "I won't get the shot because..." and expect sympathy, you'll get none...at all...especially here. No one is stepping on your dream. You are holding a personal belief higher than your dream, and that's no one's choice but yours. Pick one.
2: Decide now what precisely that 'itch' is. If it's to fly, there are absolutely no all-denying barriers to you. Serious obstacles? Yes, but nothing completely stopping you. Life will suck. Pay will be low. Hours will blow. For a while. Make up your mind that you want to fly, then find a way.
There. Are. Many. Even from a military background, I've yet to meet two guys with the same exact path. From the civil side, there are innumerable ways to end up in an airplane. It will heavily depend on where you're willing to live and what lifestyle you're willing to accept to get to your desired end state. Even then, the end state will change by the time you get there. Accept that as fact and your life will be easier.
If you are a family man, and you want to be a pilot, your family life will suffer to some degree or another. Schedules suck. You will be away from home. You will miss birthdays, anniversaries, holidays, etc. But you were wanting a military life...so you should have been ok with that, right? Because military life is like that, only worse: you're gone longer, to worse places, for less money, while leaving your family is bad locations away from family, and oh yeah, people may be actively trying to kill you.
Decide WITH YOUR FAMILY what level of life style you're willing to accept, then get to grinding. (If you exclude your bride from that decision, you will end up divorced)
D: A 'standard' civilian track can take as little as 3 years to be in the right seat of a big jet. One track: earn your CFI, teach your ass off in little airplanes, get hired by a regional/cargo feeder, then get hired by a lower tier ACMI...OR go to an aviation academy like what United is starting...OR go fly a sky tractor crop dusting...OR go fly king-airs for CBP...OR work to get hired by a fractional...OR...fly bush cargo in Anchorage...OR...the list goes on. Get to studying. As much of a dumpster fire as AirlinePilotCenter forums are, the other side of that website is pretty informational, and you can find some pretty good data in the forums if you're willing to wade through the BS.
Decide that no job is beneath you, and you'll be fine. If you choose the millennial attitude that some jobs are just too hard or too underpaid or too dirty, then yeah, your 'itch' will never be scratched, and you will be forever limited. If you choose that road, that's no one's fault but your own. If they say you need a vaccine to get the job, and you quit looking, that's your fault, not theirs.
In short: Nut up or Shut up. There is no better time to get into aviation. Decide what you will or won't do, then get to work.
P.S. I dis-recommend SE Asia. That results in a different kind of 'itch' that I've heard never goes away.