It's a little above your budget, but my 10.5" Rock River LAR-9 will shoot 1.5" to 2" five shot groups at 100 yards with hand loaded 125 gr XTP or 125 gr HAP bullets. Back that up to 50 yards and you've got 3/4" to 1" groups, and at 25 yards it's a one hole group. I suspect it will do that with pretty much any quality jacketed bullet. Even with low cost 124 gr and 147 gr plated bullets, I'm getting 5 shot groups around 4" (6" with the 124s) at 100 yards, and it'll do it all day long cold barrel or hot.
I also like Rock River's modified Colt pattern design and the ability to use RRA, Colt, Metalform or modified Uzi magazines. Surplus German and Israeli Uzi magazines are still pretty cheap with excellent condition magazines selling for around $10-$12. They are also incredibly durable.
I've had exactly one malfunction with my LAR-9 and that was ammo related. Cast bullets, FMJs hollow points - it isn't picky.
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I can't speak for other AR-9s but it is certainly possible for a well made AR-9 with a decent barrel to deliver very good accuracy. And or course, you get all the ability to customize the configuration of the rifle that you'd get with any other AR.
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My caution on 9mm ARs is that you do have to pay some attention to how you mix and match the parts if you build one. The bolt is just that - a solid bolt, no bolt and bolt carrier and it's a blow back design that requires a heavier 9mm buffer, and a compatible buffer tube and/or spacer to run right. The specific bolt carrier design (ramped or not) also determines the hammer profile that will work, with non ramped bolts normally requiring non notched or full auto hammers.