I think the critical point you've brought up is precision.
I have and have had pretty much every controller out there. IMO CH Fighter or Combat stick is what you're looking for. None of yet to surpass this stick, again IMO, when it comes to precise gunnery control. I have the Warthog, various Saiteks including the newest variant, a modded Cougar, and about dozen others I've tried over the years. I still have my original TM F22 and Suncom stuff. Again, none, be they old or new, have made me move to them from CH for AH type games. Sure for DCS and other types of sims I like the Warthog, but it just isn't as precise, or more accurately, as easy to BE precise, as the CH sticks. YMMV, but I think most who have used various sticks/hotas would feel the same, that the CH stick is lighter, more accurate, and simpler to aim with. It may be a bit TOO "loose" or whatever for some, but in a game where shooting is so important and frequent as it is in AH, it's the way to go IMO.
I would buy a CH Fighter of Combatstick for 120$ish bucks or less, and start there. If you feel you want to go with the throttle or pedals, it's only money, about 100$ish or more each for those.
2nd place I would say probably the Saitek X55 - it's fairly cheap, although there have been issues with some users, overall I would say it's not horrible quality, and would give you a Hotas all in one purchase. The movement/accuracy of the stick is again, 2nd to the CH, and around the same as the Warthog in terms of easy precision/accuracy at less than 1/2 the cost.
3rd, if $ is truly no object, a Warthog setup isn't bad. Get some custom rudder pedals, I like the Cam BF109s or Crosswinds, but there are other good options, there are brake less good custom pedals I like as well from VKB/T rudder. For generic, the Saitek Combat pedals are the best, if you mod them to remove the center detent, a PITA 30 minute job or so, lots of utube vids on this. I spent 1500$ on pedals alone last year and received much wife ack, to the point of giving up my 5960x system as a peace offering. Again, the TM Warthog will be playing second fiddle to the less expensive CH stick/hotas in terms of AH capability, but overall the craftsmanship and capabilities of the WH may make you not care.
Just my opinion/experience. Stuff to avoid - Logitech Hotas if you can even find it, as well as the force sensing Saitek stick, I returned mine it was so terrible for AH, and everything else. They can still be found, so avoid them completely IMO. x65 IRRC - booo.