I guess this is just my viewpoint...
First you have to have $14,000 laying around that you have absolutely no use for. Most people aren't fortunate enough to be in that position. Most people balancing a family and their own dreams aren't. Let's say you have the $14,000. My question is, in your scenario, what's "a few hundred a month"? $300? More? At $300 you've saved $3,600 in that first year. That's certainly not enough to complete a build in "a year or two"...you're only at $17,600! Two years at the same money you're at $21,200...perhaps you can be finished but that's the absolute most basic of builds with you doing everything. The end of year 3 at the same money you're at $24,800...a build can be done at that price, nothing fancy. This puts your time frame not at a year or two, but at 2 or 3 years with 3 years being the most realistic number. Again, that's at "a few hundred dollars" being $300...your time frame would accelerate with more money obviously.
I bring this up because you could go the above route...or enjoy your life.
I guess the biggest exception I take with your post is the portion about..."I think the car is just so much more enjoyable to drive knowing I have no payments". Maybe I'm just a simple mid-west guy with a different mindset, but I can't wrap my head around that logic. I traded two cars that I owned in order to get my first Cobra 7 years ago when I was 33 and had to finance the rest. I loved the ever living hell out of that car!!! I drove that car any and every chance I got. I went to every car show with it. I took people for a ride at gas stations when they stop and asked questions. I took neighbors for rides that I hadn't met before. I had random kids in the neighborhood come over to help me work on the car. Most importantly, I drove my daughter to her HS graduation in the car. I drove my wife's best friend to her wedding in the car and you would have thought she was the Queen of England how she was being treated as we drove down the street and the pictures that were being taken at every light.
I traded that car when my son came along for a '65 Fastback. I built and awesome GT 350 replica that I traded for a Viper and some cash which I traded 2 months later for the Cobra I have now. I don't owe anything on my current Cobra.
The point...looking back now never, ever, never would the thought have crossed my mind that "I'm having the time of my life, but this would just be so much better if I didn't owe anything on the car". I can't even fathom that thought process, there was nothing better! I loved every last second with that car. I in no way had less enjoyment because I owed money on something, what I had/have, is memories I wouldn't have had because I didn't want a loan. I'll remember driving my daughter to her graduation until the day I die. I'll remember driving Shawne to her wedding until the day I die. Hell, I'll never forget the car being dropped off the trailer at my house and my buddy and I tearing around the neighborhood in the middle of a Central Illinois December freezing to death with the biggest smiles in the world on our faces!
I guess what I'm saying is, yes, you can slowly do something over a number of years until you have all the money you need hoping that life doesn't interfere in any way OR you can enjoy your life. I've done it one way and know I'll never regret that decision.
**full disclosure...I didn't take out a 20%+ loan, if that's your option, it's a bad one, don't do it!