Skip to the bottom for the most reliable offering if you can deal with the third row in the older models.
We tried a Subaru Tribeca in 2010 (there was only one design) and with the slightly stretched Outback platform, it never worked for anything other than kids under 10 years old. So that is definately not a possiblity. The new Subaru Accent is a true three row vehicle, but is way out of your price range. So Subaru is out as the Outback mentioned only seats 5 in two rows in a squeeze.
While Lectrofuel is correct about the Toyota Sequoia you are again going to be hard pressed to stay in your price range. A 2006 with 108,000 goes for about $11,000 near me. However, like he said, with good maintenance Toyotas are still the best in reliablity although the margin with other manufactuers has shunk as of late and they have had a few cars that have issues in redesign years (not this one however). However, the 2006 model year had tranmission problems. After 2007 (2008 on) that problem went away. So that car I mention - it might be being sold for that amount because it has the chronic transmission problems of that year.
I have not had the problems of fitting people in the third row of a 2010 Buick Enclave I owned for 6 years and nearly 80,000 miles with no problems (when I could not make Subaru work for me). My dad is 6 feet tall and fit back in the third row for nearly three hours driving to and from Mt. Washington which I climbed in the car with 6 people (my mom was back there as well but she is shorter). He in fact commented that he was surprised how comfortable it was. I agree Norm is biased towards GM, but others are biased against it. I am neutral on it as some of their cars are good as they leverage Kaizen process improvement (six sigma) just like Toyota now, but sometimes their execution is suspect.
My thoughts on GM? The Enclave is the best of the bunch in reliablity but they don't last as long as the above Toyota and you would be probably happy to get 120,000 to 150,000 on one of the Lambda triplets before problems set in. Right now according to Consumer Reports 2013 on is fairly reliable for the Buick Enclave. For 2013, 2014, and 2015 the one trouble spot is the climate system, tha that is it. On average the 2013 would have 72,000 miles so I would bet you could get another 50,000 on that car.
The GMC Arcadia is an interesting car. My brother owns one he bought used for $20,000 a few years ago, and now it is six year old (a 2012). He had one problem in the initial warranty period, the side-view sensor system failed twice. When he switched dealers they fixed it four years ago and it has worked fine ever since. That has more to do with the dealer service quality than anything else. I had to convince him that GM's quality had improved to get him to buy it (he only bought Toyota before this) but he is happy with the car now that the one issue is fixed. He has 110,000 on it now and he is still happy. The only caveat I have about this brand from GM is for some reason, Consumer Reports users don't like this car - 2012 is reliable but 2012-2014 has climiate control issues (it breaks a lot) and some have suspension issues (just a few). So quality is more spotty on the GMC than the Enclave (showing the difference between divisions).
I know some on this board love Mazda and they are pretty good cars. In the case of the CX-9 you are looking at unfortunately for you, is not the most reliable of their offerings. It was redesigned in 2016 has some engine issues (water pump in particular more often than not), paint and trim issues, and in car electronic failures. The two years previous to it (2014-2015) before the redesign were better. Paint and trim and rattles are concerns, and there were a few drivetrain concerns (rough shifting) but in those years you are far more likely to be happy. I would not go any older and the years previous to that (2011-2013) there were an assortment of issues I would avoid (both the engine and drive system are suspect those years). I get my reliablity information from Consumer Reports again here.
Honda Pilot - 2009-2012 have transmission issues but they fixed it in 2013 through 2016. Redesign in 2017 the transmission started to have problems again but not as bad as the earlier design. Paint and trim (some rust) and in car electronicis issues are on the redesigned 2016. If you like this one and don't find any rust with a close check, they are good cars these days.
The car probably best for your case that is very reliable if the third row is big enough on the older models is the Toyota Highlander. It is above average in reliablity all the way back to 2001 in Consumer Reports survey. When they get that old they do have engine and transmission issues (the cars are just too old) but they are above average compared to everything else. For truly reliable Highlanders you can start with 2004 as the issues are relatiively minor from that point on. Out of all the cars mentioned, if the third row works, it is your best bet for reliable AWD without rust to replace theTaurus X AWD.