It is not impossible to achieve profitability. Seamless (what used to be every Wall Street Banker's go-to service) was profitable for a long while. Granted, they didn't have the same level of competition we see today.
I was looking for a comment mentioning Seamless. This is the best counter example. Note that these guys bootstrapped correctly from the start; as you mentioned, group buys was a great low cost/high margin way to get started.
