Flapping Sound (not a clicking) and Starting Issues

I put my car in the shop last week. What prompted me to do this was:

I would get in the car, turn the ignition and it would start but would not hold the start. I tried again and I put my foot on the gas and it STILL immediately died.

When trying once again to crank the motor it would NOT start, but instead made a flapping noise. I am saying FLAPPING NOISE, NOT CLICKING NOISE (like a bad starter).

I let it sit 5 minutes and then it would start up fine, run perfectly flawless with NO NOISES at all. Even at a stop light it would idle fine, not bumpy or anything unusual.

Took it into the shop because a local mechanic, upon me telling him what it was doing (he never HEARD) he said "sounds to me like you might have a flywheel problem. He also put the computer reader on the vehicle and it showed no issues.

I took it into the place where I bought it knowing the flywheel was part of the drive train/ power train warranty (up to 12,000 miles or 12 months. January 23, 2016 will be 12 months and I have only put on about 5,000 miles, if that and naturally, I went thru the entire song and dance with the mechanic and while still on my way back to my house (my son took me) he called saying he tried it many times and it never did it to him. He said his first thought is that it's the starter. He says, "I can replace the starter, but that is about $140. It's not under warranty because the electrical system warranty ended after you owned it 6 months". I told him I would talk to my son and get back with him in a bit. Well, I had him replace the starter. By this time you have figured out where this is going. Yep, it's doing the identical thing. I'm not a rich person and can't afford to replace parts and pay for labor just using the "theory of deduction" method this mechanic is using on me.
I hate this crap.