Where about are you when you say “downtown”? There are several barns that get regularly recommended on here that will fit most if not all of your requirements. Here is my breakdown of them:
Mechlin Farm - VERY successful program, travels often to shows, seems to have more junior riders than AAs, horses are turned out individually in paddocks for a few hours (usually 1-2 IIRC), not sure about trailer parking, 50 min to downtown
KEE Inc - travel to shows frequently, seem to have more juniors than AAs, in my experience they have been hard to get ahold of, 40 min to downtown
Baskin Farm - not sure if they travel to shows but I do not see them around, they have a lesson program with lesson horses which is mostly made up of kids, larger barn, 36 min
Dublin Farms - small farm, trailer parking I think depending on the trailer size, larger pastures, horses turned out between meals (8ish hours), derby field/ some trails, smaller program, very nice accommodating trainer (Boo Gonzales) with mostly AA riders I believe, not sure if they travel to shows but they do host schooling shows on site, 44 min
Great Griffin - larger barn with kid lesson program, travels to shows often, several different trainers available, 41 min
Kennedy Farms - very nice facility, small turnouts IIRC, variety of both kids and AAs, travels to Vermont every summer as well as other shows, 30 min
Personally, I think your best bets are KEE and Dublin for the personal feel. If you are more of a DIY person or your horse does not need to be in full training there are other places that may work. There may also be a place that works better (is closer) if you look across the river in IL. I do NOT recommend Greensfelder Stables - the turnout is essentially non-existent.