Buying a whole aircraft yourself would not get costs down.
Owning an aircraft you have the fixed costs of insurance, maintenance and hangerage/parking. A flying school has these same costs, but is able to split them over the 500+ hours their aircraft fly every year, whereas you would never be able to fly close to that.
The cheapest way to operate an aircraft is in a group/share agreement, and you could well be part of a group and be taught on that aircraft, but the other group members may not want that and insurance could be an issue.
The other option would be to buy an aircraft and then lease it to the flying school, that way they take care of most of the fixed costs so when you want to go for a lesson you only pay the variable costs, fuel, oil, engine fund etc.
The other thing to consider is that if you own the aircraft you are learning in and then it goes down for maintenance, you are stuck on the ground, where as a flying school has plenty of other aircraft to keep you in the air.