Traveling from Jordan's Queen Alia International Airport NOT to Amman

I'm not sure what you have read about transportation costs from Queen Alia airport to Amman. It is quite far from the city so, yes, taxis can be expensive. However, there is an Airport Express bus that runs 24/7 (although I believe the frequency is reduced at night) which goes to one of Amman city's many bus stations. It doesn't cost more than a couple of JOD as far as I remember.

Unfortunately, you're also unlikely to find any buses onwards to other cities at those hours. The earliest public buses on long distance routes are usually at 5-6am and even then, bus schedules are more of a "when it's full" basis rather than strictly fixed times. There are buses from operators such as JETT that are stick to better schedule, but their offices won't be open at such odd hours.

There's Madaba, half-way between Queen Alia airport and Amman that also has bus connections to other cities. However, Amman is very much the main hub for all transport and going to Madaba would mean being more restricted with your options.

I totally get what you say Amman doesn't have too many places of interest to see; I felt the same way about the city too. I would still recommend at least one day in the city though to just soak in the atmosphere. Stay near the Roman Amphitheatre (which itself I found quite underwhelming), and walk around in the markets nearby. There are some excellent eateries to grab cheap bites. If you have time, hike to the Citadel. These are the two things I'd recommend to do in Amman if you're pressed for time. Spending some time in the city just to relax rather than see any sites, and then figure out where you want to go from them there onwards.