let me make sure i have a few things straight first.
- you wrecked your bike. i have no clue why you wrecked to start off with. could have been gravel you didn't see... taking a corner too hot... trying to miss a dog... who knows. so that's not an automatic red flag.
- the fixes weren't fixes... in all kindness, this should be a lesson. when you get work done you need to make sure it is right before you pay and leave. because you can't take it back and get it fixed now that you've messed it up again.
- stinks that your strap broke. sorry to hear about that.
- you say your parents won't let you drive in downtown nashville traffic. this lets me know a few things. (1) you are under 18. if not then time to move out of the house

(2) if you're under 18 then you don't have that much street riding experience. i would guess maybe a season's worth of riding... so a few months maybe?
- you have a dream of owning a gixxer - which in N.A. is the most wrecked and stolen motorcycle.
with all of that here's what i'll say.
first, i don't think the fz6 is a beginner's bike. there isn't much difference between it and a SS. but since you already have it we need to move on.
the main difference besides the plastics is the riding position. you could get some rear-sets and clip-ons and call it a day. you have your aggressive riding position.
second, so you have a bike you don't like. YOU'RE STILL LIVING WITH YOUR PARENTS AND HAVE A BIKE! dude, i would have taken ANY bike while still under my mom's thumb. i think you just need to stop and count your blessings before worrying about getting THE bike you want.
from a purely financial standpoint i would hold onto what you have. make the mods to make it more of what you want and fix the things that need to be fixed. in a season or two you'll have more riding experience. hopefully your bike will be in better shape because you've been working on it. you'll be able to get more money for it. that's more money to put towards the dream bike. and when you get the dream bike you will have the confidence and experience to hopefully keep it shiny side up for as long as you have it.
that last paragraph is from experience. i could not afford a motorcycle so a friend gave me one. it was a touring bike that was so not my style. but i rode it for a few seasons. now i have a nice shiny fz6 and couldn't be happier. you've got time on your side. have patience.
lastly, if you haven't, you should sign up for a basic riders course. they are a great benefit, fun and cheap.
good luck with your decision. let us know what you decide to do.