I'm not sure which brand you are referring to, but just about any brand will work as long as it is unflavored and live (if it isn't refrigerated, there's a good chance it won't work). Synergy brand is really big back where I used to live. It would work fine. Try pouring it into a wider mouth container, covering it with a cloth, and letting it sit out for a few weeks. You should get a tiny little SCOBY growing on top of your tea. That scoby plus some the tea will make for excellent starter.
You have it right. As it ferments, the yeast will continue to consume the sugar, making it virtually sugar free. The taste will go from sweet to vinegar.