OK, I'm considering what I would do if I were in the OP's situation.
I recall the sage advice from someone on this forum long ago (sorry, I forget who it was):
When traveling, if a crisis occurs, apply money liberally, and don't look back. Or something like that.
I'm going to make the case that this is exactly that kind of time, and the right approach to take here. OP is clearly a "budget traveler" (otherwise they would not have a Franken-ticket from SFO to Amman). But, they are going to spend a not-insignificant amount of money to salvage their flight problem anyway, and even when they accomplish that, it's going to be a long, messy and inconvenient slog that costs them 2-3 days of their trip. There's plenty of money and time already invested. A bit more money now could quickly and cleanly resolve the biggest challenges without wrecking the next week of their trip.
I'm assuming all of the following are true:
- American/Finnair will only do what they are obligated to do: get them to ARN on the next best alternative, and I think that may deliver them there midnight Monday (after a painfully long layover in Helsinki).
- Having missed their onward (separate tickets) RyanAir flights, that $$$ is gone (unless RyanAir will give credit for later flights - I don't know but suspect they won't). The next best available option to get from ARN to AMM is not going to be cheap -- like $300 or more, and will take 10 - 18 hours (that's AFTER the ordeal in getting to ARN). Yeesh.
- There are seats available on tonight's Royal Jordanian nonstop from JFK-AMM. It's not cheap, but neither are any other options, and those other (less expensive) options would all require burning 2 - 3 days of their time in Jordan. The "cost" of those days on the trip matters, too.
I don't how long the OP's time in Jordan is. IF they are there for many days and have slack time built in at the start of the trip, then going the "long way" from ARN may be a better choice. But I think for most of us, the time we have on our trips is the most "expensive", most constrained commodity we have.
Walk-up price for RJ262 looks like about $1223. Not cheap, but IMHO at this point, it's worth it. I'd go for that. That gets them to AMM at 3:20 pm Monday, and it "fixes" what's fallen apart.
OP should try to talk his/her way into getting AA to put them on that flight first. If they won't, I'd walk over to the Royal Jordanian counter, pull out my credit card, sigh once, then wave the card, and put the crisis behind me (then I'd go have a good, stiff drink...but FFS don't miss that next flight).
Just my opinion. Wishing the OP good luck whatever they choose. Now I'm gonna go have some breakfast.
PS: @Stay-ce -- Your overall trip destinations sound great. I'm planing a trip to Malta in October so I would welcome your impressions of the place, tips etc. Feel free to PM me anytime.