Tell us what field you are flying at and I think we can make a recommendation for you.

No matter where you are, I would not recommend going to a pilot academy. You mentioned that money was an issue at one point. Pilot academies give you the same training that you get at an FBO for more money. Ultimately, pilot academies are for pilots who want to fly professionally, who either have the extra money to burn, or don't mind having a loan for an additional $20,000-$50,000, depending upon the school and programs.

A couple of recommendations. First, the Cessna 172 is the most produced aircraft ever build in the United States (over 35,000 and still being produced everyday). Before it is retired, it will be the most produced aircraft in the history of aviation. As you can see at your flight school, you have much better odds of finding a 172 than anything else. So it is in your best interest to learn the 172. There is no need to fly the 1997 or newer 172's. They are essentially the same airplane as the older 172's, they just cost more. Fly the late 70's or early 80's 172's. They didn't call that period the "golden era" of aviation for nothing! I understand that it is frustrating right now, but you will eventually learn to fly the 172 and be happy that you did. If you ever want to take passengers, the 172 is the plane to do it in. If you want to get a commercial license, a great plane to do it in is a 172RG. The 172 is common, better for your schedule (and the flight school you are at) and I recommend taking a few hours of flight and ground training to learn.

As for the instructor, it is true that you are the customer. However, I do not see much reason for you to be upset. 20 minutes is nothing. You should have a 15-20 minute briefing before you go flying, especially at the point you are at in your training. Put it this way. You can either learn on the ground before you takeoff, or you can learn the steps in the airplane while you are concentrating on flying. The airplane is a horrible learning environment. You can either learn in a quiet, free of distractions classroom, or you can learn in the airplane when you are trying to fly, its loud, and you are being charged both for the instructor and the airplane!!

I kept detailed records of my flight training. It took me 55 hours to get my private. 10 solo and 45 dual (crowded airspace). I received 80 hours of instruction. Probably more (because one of my instructors would always round down). I am a smart guy and I did almost all my studying at home. But ground training is important, and you need to do it! Otherwise it is going to take you more hours in the airplane, and that costs a lot more than sitting on the ground with an instructor. Expect them to somewhat over train you. They should. You should know as much as possible. It will make you a better pilot. It will make you a safer pilot. Did I mention it will make you better and safer?
FYI. My oral took less than 45 minutes and my checkride took 1.3 hours. I passed with flying colors, and the DE said I made the best soft field landing he had ever seen on a checkride.

To recap. Don't go to a pilot academy. They are expensive and unnecessary. Learning to fly the 172 is a great idea and I highly recommend it. It may be frustrating for a couple hours, but well worth it. Finally, ground training is very important. 20 minutes is not a long briefing, infact I would 20 minutes is typical, if not less than it should be (for important flights).
Keep us updated if things change!