I think the legality is fairly clear. I went with the "buy a certified to train in" route. I've known people to buy a experimental to train in. I've known people who built and then trained in that plane. All are legal. The only question on any of them is who is doing the maintenance, which doesn't change the question of buying/owning/building to train.
None of the above are cheaper. No way, no how. You have to know all of the details to understand that fact, but I've never heard of anyone that saved money on their training by building/buying.
You buy a plane because you want to fly it and don't care about the money. You build a plane because you want to build a plane and don't care about the money or flying the plane. You have to enjoy the building process if you want to build. Flying is the icing, not the cake.
I took just short of 120 hours to get my private (like I said, buying to train in isn't faster or cheaper). At the OP's $140 an hour, that would have been $16,800 for that amount of time. I owned the plane for 4 years (mid-1999 to Jan 2004), spent $110k, got $60k back from the sale of it, we put in around 800 hours on the plane in that 4 years. Not including all of the time and effort I put into that Cessna 172, that would put the per-hour cost of it at $50k / 800 hours = $62.50 an hour. Anyone who has been around planes for any length of time will tell you that 200 hours a year is a TON of hours. Most small planes don't fly a quarter of that a year. I had a partner that I was very compatible with, and we both commuted with the plane to visit family, and went out of our way to use the plane as much as possible. Still came out to $62.50 an hour.
Owning a plane is never cheaper. You will not train faster in a plane you own vs. rent. You own a plane because you want to own and don't care about the money (or, you want to own a plane more than you want to own money). You build a plane because you like building, not because you want to fly.
These are facts that prove themselves over and over again. It's no big deal. It's only money. But, ask anyone who's been around small plane aviation for a length of time and you'll get the same answer.
Having said all that. I loved owning my plane. Had I not lost my medical, I'd still own it, and still be flying it.